Properties

Label 24-74e12-1.1-c1e12-0-0
Degree $24$
Conductor $2.696\times 10^{22}$
Sign $1$
Analytic cond. $0.00181178$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 6·5-s + 6·7-s − 2·8-s + 3·9-s + 3·11-s − 6·13-s + 18·15-s − 3·17-s − 3·19-s − 18·21-s − 21·23-s + 6·24-s + 15·25-s + 27-s + 6·29-s + 42·31-s − 9·33-s − 36·35-s − 3·37-s + 18·39-s + 12·40-s − 21·41-s + 36·43-s − 18·45-s + 9·47-s + 12·49-s + ⋯
L(s)  = 1  − 1.73·3-s − 2.68·5-s + 2.26·7-s − 0.707·8-s + 9-s + 0.904·11-s − 1.66·13-s + 4.64·15-s − 0.727·17-s − 0.688·19-s − 3.92·21-s − 4.37·23-s + 1.22·24-s + 3·25-s + 0.192·27-s + 1.11·29-s + 7.54·31-s − 1.56·33-s − 6.08·35-s − 0.493·37-s + 2.88·39-s + 1.89·40-s − 3.27·41-s + 5.48·43-s − 2.68·45-s + 1.31·47-s + 12/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(0.00181178\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{74} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 37^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1333055296\)
\(L(\frac12)\) \(\approx\) \(0.1333055296\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( 1 + 3 T + 30 T^{2} + 76 T^{3} + 405 T^{4} - 1863 T^{5} - 55623 T^{6} - 1863 p T^{7} + 405 p^{2} T^{8} + 76 p^{3} T^{9} + 30 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
good3 \( 1 + p T + 2 p T^{2} + 8 T^{3} + 5 p T^{4} + 7 p^{2} T^{5} + 164 T^{6} + 32 p^{2} T^{7} + 155 p T^{8} + 748 T^{9} + 563 p T^{10} + 1172 p T^{11} + 6265 T^{12} + 1172 p^{2} T^{13} + 563 p^{3} T^{14} + 748 p^{3} T^{15} + 155 p^{5} T^{16} + 32 p^{7} T^{17} + 164 p^{6} T^{18} + 7 p^{9} T^{19} + 5 p^{9} T^{20} + 8 p^{9} T^{21} + 2 p^{11} T^{22} + p^{12} T^{23} + p^{12} T^{24} \)
5 \( ( 1 + 3 T + 6 T^{2} + 8 T^{3} + 33 T^{4} + 117 T^{5} + 281 T^{6} + 117 p T^{7} + 33 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
7 \( 1 - 6 T + 24 T^{2} - 95 T^{3} + 465 T^{4} - 1857 T^{5} + 6222 T^{6} - 20469 T^{7} + 68997 T^{8} - 217108 T^{9} + 623313 T^{10} - 1758375 T^{11} + 4764887 T^{12} - 1758375 p T^{13} + 623313 p^{2} T^{14} - 217108 p^{3} T^{15} + 68997 p^{4} T^{16} - 20469 p^{5} T^{17} + 6222 p^{6} T^{18} - 1857 p^{7} T^{19} + 465 p^{8} T^{20} - 95 p^{9} T^{21} + 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 3 T - 3 p T^{2} + 130 T^{3} + 489 T^{4} - 2535 T^{5} - 4481 T^{6} + 32709 T^{7} + 25248 T^{8} - 290746 T^{9} - 45993 T^{10} + 1203945 T^{11} - 338433 T^{12} + 1203945 p T^{13} - 45993 p^{2} T^{14} - 290746 p^{3} T^{15} + 25248 p^{4} T^{16} + 32709 p^{5} T^{17} - 4481 p^{6} T^{18} - 2535 p^{7} T^{19} + 489 p^{8} T^{20} + 130 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 + 6 T - 24 T^{2} - 173 T^{3} + 363 T^{4} + 2607 T^{5} - 3158 T^{6} - 13581 T^{7} + 5157 p T^{8} - 128432 T^{9} - 2125101 T^{10} + 1146981 T^{11} + 35353633 T^{12} + 1146981 p T^{13} - 2125101 p^{2} T^{14} - 128432 p^{3} T^{15} + 5157 p^{5} T^{16} - 13581 p^{5} T^{17} - 3158 p^{6} T^{18} + 2607 p^{7} T^{19} + 363 p^{8} T^{20} - 173 p^{9} T^{21} - 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + 3 T + 18 T^{2} + 6 T^{3} - 9 p T^{4} - 1947 T^{5} - 3044 T^{6} - 9072 T^{7} + 146457 T^{8} + 298428 T^{9} + 1782945 T^{10} - 3618000 T^{11} - 11940819 T^{12} - 3618000 p T^{13} + 1782945 p^{2} T^{14} + 298428 p^{3} T^{15} + 146457 p^{4} T^{16} - 9072 p^{5} T^{17} - 3044 p^{6} T^{18} - 1947 p^{7} T^{19} - 9 p^{9} T^{20} + 6 p^{9} T^{21} + 18 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 3 T - 9 T^{2} - 9 p T^{3} - 180 T^{4} + 78 T^{5} + 4230 T^{6} + 15621 T^{7} + 197667 T^{8} + 445248 T^{9} - 684225 T^{10} - 11322285 T^{11} - 52854949 T^{12} - 11322285 p T^{13} - 684225 p^{2} T^{14} + 445248 p^{3} T^{15} + 197667 p^{4} T^{16} + 15621 p^{5} T^{17} + 4230 p^{6} T^{18} + 78 p^{7} T^{19} - 180 p^{8} T^{20} - 9 p^{10} T^{21} - 9 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 21 T + 159 T^{2} + 422 T^{3} + 579 T^{4} + 19227 T^{5} + 175277 T^{6} + 558405 T^{7} + 1422558 T^{8} + 19927232 T^{9} + 147659307 T^{10} + 402027663 T^{11} + 529096303 T^{12} + 402027663 p T^{13} + 147659307 p^{2} T^{14} + 19927232 p^{3} T^{15} + 1422558 p^{4} T^{16} + 558405 p^{5} T^{17} + 175277 p^{6} T^{18} + 19227 p^{7} T^{19} + 579 p^{8} T^{20} + 422 p^{9} T^{21} + 159 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 6 T - 114 T^{2} + 672 T^{3} + 7644 T^{4} - 39144 T^{5} - 391478 T^{6} + 1354554 T^{7} + 17795898 T^{8} - 30149472 T^{9} - 698927454 T^{10} + 306162306 T^{11} + 22713452187 T^{12} + 306162306 p T^{13} - 698927454 p^{2} T^{14} - 30149472 p^{3} T^{15} + 17795898 p^{4} T^{16} + 1354554 p^{5} T^{17} - 391478 p^{6} T^{18} - 39144 p^{7} T^{19} + 7644 p^{8} T^{20} + 672 p^{9} T^{21} - 114 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
31 \( ( 1 - 21 T + 210 T^{2} - 1538 T^{3} + 11139 T^{4} - 78321 T^{5} + 476692 T^{6} - 78321 p T^{7} + 11139 p^{2} T^{8} - 1538 p^{3} T^{9} + 210 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 + 21 T + 216 T^{2} + 1166 T^{3} - 213 T^{4} - 46413 T^{5} - 301238 T^{6} - 683154 T^{7} + 860643 T^{8} - 15398906 T^{9} - 189077871 T^{10} - 216030306 T^{11} + 2816578149 T^{12} - 216030306 p T^{13} - 189077871 p^{2} T^{14} - 15398906 p^{3} T^{15} + 860643 p^{4} T^{16} - 683154 p^{5} T^{17} - 301238 p^{6} T^{18} - 46413 p^{7} T^{19} - 213 p^{8} T^{20} + 1166 p^{9} T^{21} + 216 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \)
43 \( ( 1 - 18 T + 309 T^{2} - 3231 T^{3} + 32763 T^{4} - 247509 T^{5} + 1842654 T^{6} - 247509 p T^{7} + 32763 p^{2} T^{8} - 3231 p^{3} T^{9} + 309 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 - 9 T - 60 T^{2} + 911 T^{3} - 2808 T^{4} - 5241 T^{5} + 46456 T^{6} - 856389 T^{7} + 16637952 T^{8} - 103116317 T^{9} - 177908364 T^{10} + 4898549355 T^{11} - 33022423026 T^{12} + 4898549355 p T^{13} - 177908364 p^{2} T^{14} - 103116317 p^{3} T^{15} + 16637952 p^{4} T^{16} - 856389 p^{5} T^{17} + 46456 p^{6} T^{18} - 5241 p^{7} T^{19} - 2808 p^{8} T^{20} + 911 p^{9} T^{21} - 60 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 6 T + 6 T^{2} - 845 T^{3} - 4311 T^{4} - 4989 T^{5} + 280438 T^{6} + 1000107 T^{7} - 860991 T^{8} - 81100084 T^{9} - 161986593 T^{10} + 1514513289 T^{11} + 36337260753 T^{12} + 1514513289 p T^{13} - 161986593 p^{2} T^{14} - 81100084 p^{3} T^{15} - 860991 p^{4} T^{16} + 1000107 p^{5} T^{17} + 280438 p^{6} T^{18} - 4989 p^{7} T^{19} - 4311 p^{8} T^{20} - 845 p^{9} T^{21} + 6 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 6 T + 12 T^{2} - 1145 T^{3} - 4431 T^{4} + 447 p T^{5} + 836518 T^{6} + 2024073 T^{7} - 25270923 T^{8} - 375375964 T^{9} - 906247443 T^{10} + 10995722067 T^{11} + 151700550339 T^{12} + 10995722067 p T^{13} - 906247443 p^{2} T^{14} - 375375964 p^{3} T^{15} - 25270923 p^{4} T^{16} + 2024073 p^{5} T^{17} + 836518 p^{6} T^{18} + 447 p^{8} T^{19} - 4431 p^{8} T^{20} - 1145 p^{9} T^{21} + 12 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 18 T + 363 T^{2} + 4801 T^{3} + 61101 T^{4} + 653133 T^{5} + 6571197 T^{6} + 59319195 T^{7} + 503874870 T^{8} + 4054572020 T^{9} + 31129024686 T^{10} + 242279892813 T^{11} + 1846104745820 T^{12} + 242279892813 p T^{13} + 31129024686 p^{2} T^{14} + 4054572020 p^{3} T^{15} + 503874870 p^{4} T^{16} + 59319195 p^{5} T^{17} + 6571197 p^{6} T^{18} + 653133 p^{7} T^{19} + 61101 p^{8} T^{20} + 4801 p^{9} T^{21} + 363 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 27 T + 444 T^{2} + 5436 T^{3} + 53838 T^{4} + 473337 T^{5} + 3670158 T^{6} + 23329314 T^{7} + 87419250 T^{8} - 321151680 T^{9} - 10037484510 T^{10} - 120618475362 T^{11} - 1073964440161 T^{12} - 120618475362 p T^{13} - 10037484510 p^{2} T^{14} - 321151680 p^{3} T^{15} + 87419250 p^{4} T^{16} + 23329314 p^{5} T^{17} + 3670158 p^{6} T^{18} + 473337 p^{7} T^{19} + 53838 p^{8} T^{20} + 5436 p^{9} T^{21} + 444 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 18 T + 192 T^{2} + 1440 T^{3} + 6492 T^{4} - 17532 T^{5} - 666482 T^{6} - 6524802 T^{7} - 58934700 T^{8} - 311617008 T^{9} + 1772614548 T^{10} + 631778814 p T^{11} + 494416409199 T^{12} + 631778814 p^{2} T^{13} + 1772614548 p^{2} T^{14} - 311617008 p^{3} T^{15} - 58934700 p^{4} T^{16} - 6524802 p^{5} T^{17} - 666482 p^{6} T^{18} - 17532 p^{7} T^{19} + 6492 p^{8} T^{20} + 1440 p^{9} T^{21} + 192 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
73 \( ( 1 - 27 T + 546 T^{2} - 7362 T^{3} + 89157 T^{4} - 878589 T^{5} + 8191977 T^{6} - 878589 p T^{7} + 89157 p^{2} T^{8} - 7362 p^{3} T^{9} + 546 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( 1 + 12 T - 54 T^{2} - 1860 T^{3} - 13086 T^{4} + 137874 T^{5} + 2485510 T^{6} + 7128018 T^{7} - 102763476 T^{8} - 1697189328 T^{9} - 5386377564 T^{10} + 87929254494 T^{11} + 1098603446871 T^{12} + 87929254494 p T^{13} - 5386377564 p^{2} T^{14} - 1697189328 p^{3} T^{15} - 102763476 p^{4} T^{16} + 7128018 p^{5} T^{17} + 2485510 p^{6} T^{18} + 137874 p^{7} T^{19} - 13086 p^{8} T^{20} - 1860 p^{9} T^{21} - 54 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 6 T - 24 T^{2} - 2009 T^{3} - 10335 T^{4} + 24429 T^{5} + 1757830 T^{6} + 3624057 T^{7} - 15147003 T^{8} - 898429792 T^{9} + 1619776653 T^{10} - 5352801765 T^{11} + 500484873171 T^{12} - 5352801765 p T^{13} + 1619776653 p^{2} T^{14} - 898429792 p^{3} T^{15} - 15147003 p^{4} T^{16} + 3624057 p^{5} T^{17} + 1757830 p^{6} T^{18} + 24429 p^{7} T^{19} - 10335 p^{8} T^{20} - 2009 p^{9} T^{21} - 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 15 T - 3 T^{2} - 1194 T^{3} - 3846 T^{4} + 128631 T^{5} + 1637938 T^{6} - 2093373 T^{7} - 150612822 T^{8} - 858295905 T^{9} - 430305669 T^{10} - 7812564102 T^{11} - 274651573950 T^{12} - 7812564102 p T^{13} - 430305669 p^{2} T^{14} - 858295905 p^{3} T^{15} - 150612822 p^{4} T^{16} - 2093373 p^{5} T^{17} + 1637938 p^{6} T^{18} + 128631 p^{7} T^{19} - 3846 p^{8} T^{20} - 1194 p^{9} T^{21} - 3 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 42 T + 633 T^{2} + 2432 T^{3} - 22086 T^{4} + 261732 T^{5} + 13128348 T^{6} + 161653860 T^{7} + 855575061 T^{8} - 675970844 T^{9} + 7426115355 T^{10} + 1410680549910 T^{11} + 22432258880033 T^{12} + 1410680549910 p T^{13} + 7426115355 p^{2} T^{14} - 675970844 p^{3} T^{15} + 855575061 p^{4} T^{16} + 161653860 p^{5} T^{17} + 13128348 p^{6} T^{18} + 261732 p^{7} T^{19} - 22086 p^{8} T^{20} + 2432 p^{9} T^{21} + 633 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.62298570593166065922861843854, −5.45694800639675010816529130081, −5.10691019732322619661000002747, −4.97338589124328749649671196315, −4.89447097243458177858122036153, −4.71995734231250318682238379607, −4.64619723666353755724299713111, −4.51575352084716419805689197947, −4.46444877350296453794281036012, −4.37859637584662036455254527585, −4.21244474004040785437106630639, −4.10194151101519860627686266291, −4.01169869595108353775807037165, −3.98282362841921112276334608742, −3.64938138511806075628993087682, −3.15439325335823151430107622770, −3.02637810980931561368589511780, −3.01016367370750207830118677858, −3.00192275323802422962134276611, −2.54006356054050893070693495172, −2.31994716054009641416243844545, −2.19179960255726958692821388548, −1.82942286564869793950187444219, −1.48628071627190231624132091261, −0.815260982004608777935867505137, 0.815260982004608777935867505137, 1.48628071627190231624132091261, 1.82942286564869793950187444219, 2.19179960255726958692821388548, 2.31994716054009641416243844545, 2.54006356054050893070693495172, 3.00192275323802422962134276611, 3.01016367370750207830118677858, 3.02637810980931561368589511780, 3.15439325335823151430107622770, 3.64938138511806075628993087682, 3.98282362841921112276334608742, 4.01169869595108353775807037165, 4.10194151101519860627686266291, 4.21244474004040785437106630639, 4.37859637584662036455254527585, 4.46444877350296453794281036012, 4.51575352084716419805689197947, 4.64619723666353755724299713111, 4.71995734231250318682238379607, 4.89447097243458177858122036153, 4.97338589124328749649671196315, 5.10691019732322619661000002747, 5.45694800639675010816529130081, 5.62298570593166065922861843854

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.