Properties

Label 24-637e12-1.1-c1e12-0-5
Degree $24$
Conductor $4.463\times 10^{33}$
Sign $1$
Analytic cond. $2.99915\times 10^{8}$
Root an. cond. $2.25532$
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 − 4·4-s + 13·9-s − 12·11-s − 12·12-s + 2·13-s + 8·16-s − 17·17-s + 9·19-s + 3·23-s + 35·25-s + 32·27-s − 29-s − 36·33-s − 52·36-s − 15·37-s + 6·39-s + 6·41-s + 11·43-s + 48·44-s + 24·48-s − 51·51-s − 8·52-s + 16·53-s + 27·57-s + 27·59-s − 5·61-s + ⋯
L(s)  = 1  + 1.73·3-s − 2·4-s + 13/3·9-s − 3.61·11-s − 3.46·12-s + 0.554·13-s + 2·16-s − 4.12·17-s + 2.06·19-s + 0.625·23-s + 7·25-s + 6.15·27-s − 0.185·29-s − 6.26·33-s − 8.66·36-s − 2.46·37-s + 0.960·39-s + 0.937·41-s + 1.67·43-s + 7.23·44-s + 3.46·48-s − 7.14·51-s − 1.10·52-s + 2.19·53-s + 3.57·57-s + 3.51·59-s − 0.640·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{24} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(2.99915\times 10^{8}\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.464344205\)
\(L(\frac12)\) \(\approx\) \(2.464344205\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 - 2 T - 18 T^{2} + 17 T^{3} + 341 T^{4} + 63 T^{5} - 6395 T^{6} + 63 p T^{7} + 341 p^{2} T^{8} + 17 p^{3} T^{9} - 18 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
good2 \( 1 + p^{2} T^{2} + p^{3} T^{4} + 5 T^{6} - p^{4} T^{8} - 3 p^{4} T^{10} + 3 p^{2} T^{11} - 111 T^{12} + 3 p^{3} T^{13} - 3 p^{6} T^{14} - p^{8} T^{16} + 5 p^{6} T^{18} + p^{11} T^{20} + p^{12} T^{22} + p^{12} T^{24} \)
3 \( 1 - p T - 4 T^{2} + 19 T^{3} + p T^{4} - 16 p T^{5} - 19 T^{6} + 47 p T^{7} + 22 T^{8} - 73 p T^{9} - 95 p T^{10} - 124 T^{11} + 2353 T^{12} - 124 p T^{13} - 95 p^{3} T^{14} - 73 p^{4} T^{15} + 22 p^{4} T^{16} + 47 p^{6} T^{17} - 19 p^{6} T^{18} - 16 p^{8} T^{19} + p^{9} T^{20} + 19 p^{9} T^{21} - 4 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \)
5 \( 1 - 7 p T^{2} + 601 T^{4} - 6779 T^{6} + 56873 T^{8} - 379271 T^{10} + 2080181 T^{12} - 379271 p^{2} T^{14} + 56873 p^{4} T^{16} - 6779 p^{6} T^{18} + 601 p^{8} T^{20} - 7 p^{11} T^{22} + p^{12} T^{24} \)
11 \( 1 + 12 T + 107 T^{2} + 708 T^{3} + 3961 T^{4} + 18831 T^{5} + 79826 T^{6} + 305121 T^{7} + 98539 p T^{8} + 3659247 T^{9} + 11994110 T^{10} + 39258729 T^{11} + 129081947 T^{12} + 39258729 p T^{13} + 11994110 p^{2} T^{14} + 3659247 p^{3} T^{15} + 98539 p^{5} T^{16} + 305121 p^{5} T^{17} + 79826 p^{6} T^{18} + 18831 p^{7} T^{19} + 3961 p^{8} T^{20} + 708 p^{9} T^{21} + 107 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + p T + 91 T^{2} + 80 T^{3} + 354 T^{4} + 13150 T^{5} + 66387 T^{6} + 80838 T^{7} + 350534 T^{8} + 5285257 T^{9} + 24796664 T^{10} + 64859774 T^{11} + 191483227 T^{12} + 64859774 p T^{13} + 24796664 p^{2} T^{14} + 5285257 p^{3} T^{15} + 350534 p^{4} T^{16} + 80838 p^{5} T^{17} + 66387 p^{6} T^{18} + 13150 p^{7} T^{19} + 354 p^{8} T^{20} + 80 p^{9} T^{21} + 91 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \)
19 \( 1 - 9 T + 115 T^{2} - 792 T^{3} + 6216 T^{4} - 34521 T^{5} + 202721 T^{6} - 930966 T^{7} + 4436762 T^{8} - 17205552 T^{9} + 72759135 T^{10} - 265082310 T^{11} + 1203810589 T^{12} - 265082310 p T^{13} + 72759135 p^{2} T^{14} - 17205552 p^{3} T^{15} + 4436762 p^{4} T^{16} - 930966 p^{5} T^{17} + 202721 p^{6} T^{18} - 34521 p^{7} T^{19} + 6216 p^{8} T^{20} - 792 p^{9} T^{21} + 115 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 3 T - 79 T^{2} - 4 p T^{3} + 4336 T^{4} + 12687 T^{5} - 111893 T^{6} - 708984 T^{7} + 1416466 T^{8} + 17335200 T^{9} + 27721271 T^{10} - 199047772 T^{11} - 1123052895 T^{12} - 199047772 p T^{13} + 27721271 p^{2} T^{14} + 17335200 p^{3} T^{15} + 1416466 p^{4} T^{16} - 708984 p^{5} T^{17} - 111893 p^{6} T^{18} + 12687 p^{7} T^{19} + 4336 p^{8} T^{20} - 4 p^{10} T^{21} - 79 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 + T - 3 p T^{2} + 178 T^{3} + 128 p T^{4} - 15194 T^{5} - 71197 T^{6} + 410532 T^{7} + 447076 T^{8} - 2082223 T^{9} - 28038240 T^{10} - 61539492 T^{11} + 1484469159 T^{12} - 61539492 p T^{13} - 28038240 p^{2} T^{14} - 2082223 p^{3} T^{15} + 447076 p^{4} T^{16} + 410532 p^{5} T^{17} - 71197 p^{6} T^{18} - 15194 p^{7} T^{19} + 128 p^{9} T^{20} + 178 p^{9} T^{21} - 3 p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 140 T^{2} + 9930 T^{4} - 450715 T^{6} + 15153149 T^{8} - 429707637 T^{10} + 12513324193 T^{12} - 429707637 p^{2} T^{14} + 15153149 p^{4} T^{16} - 450715 p^{6} T^{18} + 9930 p^{8} T^{20} - 140 p^{10} T^{22} + p^{12} T^{24} \)
37 \( 1 + 15 T + 261 T^{2} + 2790 T^{3} + 30330 T^{4} + 259743 T^{5} + 2192045 T^{6} + 15890796 T^{7} + 113579334 T^{8} + 728456778 T^{9} + 4731329259 T^{10} + 28450921716 T^{11} + 178599894873 T^{12} + 28450921716 p T^{13} + 4731329259 p^{2} T^{14} + 728456778 p^{3} T^{15} + 113579334 p^{4} T^{16} + 15890796 p^{5} T^{17} + 2192045 p^{6} T^{18} + 259743 p^{7} T^{19} + 30330 p^{8} T^{20} + 2790 p^{9} T^{21} + 261 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 6 T + 87 T^{2} - 450 T^{3} + 3253 T^{4} - 22749 T^{5} + 119000 T^{6} - 655563 T^{7} + 2153315 T^{8} + 1902885 T^{9} - 21527678 T^{10} + 258610773 T^{11} - 1564835407 T^{12} + 258610773 p T^{13} - 21527678 p^{2} T^{14} + 1902885 p^{3} T^{15} + 2153315 p^{4} T^{16} - 655563 p^{5} T^{17} + 119000 p^{6} T^{18} - 22749 p^{7} T^{19} + 3253 p^{8} T^{20} - 450 p^{9} T^{21} + 87 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 11 T - 88 T^{2} + 799 T^{3} + 8712 T^{4} - 35151 T^{5} - 615381 T^{6} + 1493311 T^{7} + 26239709 T^{8} - 29370632 T^{9} - 1062722582 T^{10} - 167088622 T^{11} + 50469301069 T^{12} - 167088622 p T^{13} - 1062722582 p^{2} T^{14} - 29370632 p^{3} T^{15} + 26239709 p^{4} T^{16} + 1493311 p^{5} T^{17} - 615381 p^{6} T^{18} - 35151 p^{7} T^{19} + 8712 p^{8} T^{20} + 799 p^{9} T^{21} - 88 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - 523 T^{2} + 127033 T^{4} - 18954871 T^{6} + 1935563261 T^{8} - 142521847195 T^{10} + 7760213706737 T^{12} - 142521847195 p^{2} T^{14} + 1935563261 p^{4} T^{16} - 18954871 p^{6} T^{18} + 127033 p^{8} T^{20} - 523 p^{10} T^{22} + p^{12} T^{24} \)
53 \( ( 1 - 8 T + 280 T^{2} - 1716 T^{3} + 33468 T^{4} - 160748 T^{5} + 2272305 T^{6} - 160748 p T^{7} + 33468 p^{2} T^{8} - 1716 p^{3} T^{9} + 280 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 - 27 T + 442 T^{2} - 5373 T^{3} + 47243 T^{4} - 314460 T^{5} + 1356131 T^{6} - 82953 T^{7} - 42334024 T^{8} + 284989455 T^{9} + 1152047535 T^{10} - 39755713086 T^{11} + 391751656497 T^{12} - 39755713086 p T^{13} + 1152047535 p^{2} T^{14} + 284989455 p^{3} T^{15} - 42334024 p^{4} T^{16} - 82953 p^{5} T^{17} + 1356131 p^{6} T^{18} - 314460 p^{7} T^{19} + 47243 p^{8} T^{20} - 5373 p^{9} T^{21} + 442 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 5 T - 266 T^{2} - 887 T^{3} + 40211 T^{4} + 80704 T^{5} - 4493895 T^{6} - 5470553 T^{7} + 403359860 T^{8} + 265514337 T^{9} - 30729415795 T^{10} - 6069628418 T^{11} + 2020503763113 T^{12} - 6069628418 p T^{13} - 30729415795 p^{2} T^{14} + 265514337 p^{3} T^{15} + 403359860 p^{4} T^{16} - 5470553 p^{5} T^{17} - 4493895 p^{6} T^{18} + 80704 p^{7} T^{19} + 40211 p^{8} T^{20} - 887 p^{9} T^{21} - 266 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 15 T + 295 T^{2} + 3300 T^{3} + 37385 T^{4} + 337086 T^{5} + 2917472 T^{6} + 22840887 T^{7} + 168746996 T^{8} + 1188097515 T^{9} + 6974328225 T^{10} + 52016179137 T^{11} + 290088466863 T^{12} + 52016179137 p T^{13} + 6974328225 p^{2} T^{14} + 1188097515 p^{3} T^{15} + 168746996 p^{4} T^{16} + 22840887 p^{5} T^{17} + 2917472 p^{6} T^{18} + 337086 p^{7} T^{19} + 37385 p^{8} T^{20} + 3300 p^{9} T^{21} + 295 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 30 T + 653 T^{2} - 10590 T^{3} + 145585 T^{4} - 1771113 T^{5} + 19362440 T^{6} - 197718039 T^{7} + 1886667311 T^{8} - 17315970285 T^{9} + 153782228042 T^{10} - 1330750537803 T^{11} + 11347613768747 T^{12} - 1330750537803 p T^{13} + 153782228042 p^{2} T^{14} - 17315970285 p^{3} T^{15} + 1886667311 p^{4} T^{16} - 197718039 p^{5} T^{17} + 19362440 p^{6} T^{18} - 1771113 p^{7} T^{19} + 145585 p^{8} T^{20} - 10590 p^{9} T^{21} + 653 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 476 T^{2} + 113546 T^{4} - 17894407 T^{6} + 2099846477 T^{8} - 197467415277 T^{10} + 15594902689281 T^{12} - 197467415277 p^{2} T^{14} + 2099846477 p^{4} T^{16} - 17894407 p^{6} T^{18} + 113546 p^{8} T^{20} - 476 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 35 T + 818 T^{2} - 13321 T^{3} + 181646 T^{4} - 2028165 T^{5} + 19730991 T^{6} - 2028165 p T^{7} + 181646 p^{2} T^{8} - 13321 p^{3} T^{9} + 818 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 533 T^{2} + 130220 T^{4} - 19632136 T^{6} + 2123453318 T^{8} - 188332543809 T^{10} + 15642556571895 T^{12} - 188332543809 p^{2} T^{14} + 2123453318 p^{4} T^{16} - 19632136 p^{6} T^{18} + 130220 p^{8} T^{20} - 533 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 - 48 T + 1471 T^{2} - 33744 T^{3} + 649840 T^{4} - 10865652 T^{5} + 162607993 T^{6} - 2211532644 T^{7} + 27724965467 T^{8} - 323293945128 T^{9} + 3530567707708 T^{10} - 36297047904324 T^{11} + 352202680354553 T^{12} - 36297047904324 p T^{13} + 3530567707708 p^{2} T^{14} - 323293945128 p^{3} T^{15} + 27724965467 p^{4} T^{16} - 2211532644 p^{5} T^{17} + 162607993 p^{6} T^{18} - 10865652 p^{7} T^{19} + 649840 p^{8} T^{20} - 33744 p^{9} T^{21} + 1471 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 3 T + 406 T^{2} - 1209 T^{3} + 93212 T^{4} - 404427 T^{5} + 13886771 T^{6} - 91101243 T^{7} + 1491993611 T^{8} - 15578221158 T^{9} + 128844085584 T^{10} - 1984216560294 T^{11} + 11260776987123 T^{12} - 1984216560294 p T^{13} + 128844085584 p^{2} T^{14} - 15578221158 p^{3} T^{15} + 1491993611 p^{4} T^{16} - 91101243 p^{5} T^{17} + 13886771 p^{6} T^{18} - 404427 p^{7} T^{19} + 93212 p^{8} T^{20} - 1209 p^{9} T^{21} + 406 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
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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

−3.53085791453739003672187038730, −3.35812467213061406444756666927, −3.27780210006039101593507827785, −3.25521062628946034535734233028, −3.15236968530946492173716177039, −2.68852882854303297015162879193, −2.66860435980687688985882495212, −2.63016568604490413660556673652, −2.59969487735756121153742345515, −2.48932704501639980719271345451, −2.47153750208147208769271121449, −2.42525198740897390949731410279, −2.33163009135923955540045653409, −2.31058297831586180100808012332, −2.09905615611207210115405736401, −1.73534964415087999667675464087, −1.63466827974243403840762533451, −1.40321168277685861340663715578, −1.26006995341197000089899129657, −1.10014001918704500223423609720, −1.06805285349097292043703720864, −0.988370147811691151376066508461, −0.73256520655274577548863782401, −0.56897358995766931228504353134, −0.12850407696643018076043158683, 0.12850407696643018076043158683, 0.56897358995766931228504353134, 0.73256520655274577548863782401, 0.988370147811691151376066508461, 1.06805285349097292043703720864, 1.10014001918704500223423609720, 1.26006995341197000089899129657, 1.40321168277685861340663715578, 1.63466827974243403840762533451, 1.73534964415087999667675464087, 2.09905615611207210115405736401, 2.31058297831586180100808012332, 2.33163009135923955540045653409, 2.42525198740897390949731410279, 2.47153750208147208769271121449, 2.48932704501639980719271345451, 2.59969487735756121153742345515, 2.63016568604490413660556673652, 2.66860435980687688985882495212, 2.68852882854303297015162879193, 3.15236968530946492173716177039, 3.25521062628946034535734233028, 3.27780210006039101593507827785, 3.35812467213061406444756666927, 3.53085791453739003672187038730

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

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