Properties

Label 24-637e12-1.1-c1e12-0-15
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 + 8·4-s + 3·5-s + 13·9-s + 12·11-s + 24·12-s + 2·13-s + 9·15-s + 32·16-s + 34·17-s − 9·19-s + 24·20-s − 6·23-s − 13·25-s + 32·27-s − 29-s − 18·31-s + 36·33-s + 104·36-s + 6·39-s + 6·41-s + 11·43-s + 96·44-s + 39·45-s + 15·47-s + 96·48-s + 102·51-s + ⋯
L(s)  = 1  + 1.73·3-s + 4·4-s + 1.34·5-s + 13/3·9-s + 3.61·11-s + 6.92·12-s + 0.554·13-s + 2.32·15-s + 8·16-s + 8.24·17-s − 2.06·19-s + 5.36·20-s − 1.25·23-s − 2.59·25-s + 6.15·27-s − 0.185·29-s − 3.23·31-s + 6.26·33-s + 52/3·36-s + 0.960·39-s + 0.937·41-s + 1.67·43-s + 14.4·44-s + 5.81·45-s + 2.18·47-s + 13.8·48-s + 14.2·51-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

\(L(1)\) \(\approx\) \(513.0362294\)
\(L(\frac12)\) \(\approx\) \(513.0362294\)
\(L(\frac{3}{2})\) 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 \( 1 - p^{3} T^{2} + p^{5} T^{4} - 91 T^{6} + 7 p^{5} T^{8} - 33 p^{4} T^{10} + 1137 T^{12} - 33 p^{6} T^{14} + 7 p^{9} T^{16} - 91 p^{6} T^{18} + p^{13} T^{20} - p^{13} T^{22} + p^{12} T^{24} \)
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 \( 1 - 3 T + 22 T^{2} - 57 T^{3} + 241 T^{4} - 18 p^{2} T^{5} + 1411 T^{6} - 1311 T^{7} + 3032 T^{8} + 7749 T^{9} - 3571 p T^{10} + 104772 T^{11} - 169399 T^{12} + 104772 p T^{13} - 3571 p^{3} T^{14} + 7749 p^{3} T^{15} + 3032 p^{4} T^{16} - 1311 p^{5} T^{17} + 1411 p^{6} T^{18} - 18 p^{9} T^{19} + 241 p^{8} T^{20} - 57 p^{9} T^{21} + 22 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
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 \( ( 1 - p T + 198 T^{2} - 1643 T^{3} + 10919 T^{4} - 59082 T^{5} + 266647 T^{6} - 59082 p T^{7} + 10919 p^{2} T^{8} - 1643 p^{3} T^{9} + 198 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \)
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 \( ( 1 + 3 T + 88 T^{2} + 86 T^{3} + 3150 T^{4} - 1355 T^{5} + 76923 T^{6} - 1355 p T^{7} + 3150 p^{2} T^{8} + 86 p^{3} T^{9} + 88 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 + 18 T + 232 T^{2} + 72 p T^{3} + 18099 T^{4} + 118611 T^{5} + 671327 T^{6} + 3244986 T^{7} + 13790561 T^{8} + 53577873 T^{9} + 222950535 T^{10} + 1049414532 T^{11} + 5476860865 T^{12} + 1049414532 p T^{13} + 222950535 p^{2} T^{14} + 53577873 p^{3} T^{15} + 13790561 p^{4} T^{16} + 3244986 p^{5} T^{17} + 671327 p^{6} T^{18} + 118611 p^{7} T^{19} + 18099 p^{8} T^{20} + 72 p^{10} T^{21} + 232 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 297 T^{2} + 43128 T^{4} - 4084528 T^{6} + 281694330 T^{8} - 14898657933 T^{10} + 619410154695 T^{12} - 14898657933 p^{2} T^{14} + 281694330 p^{4} T^{16} - 4084528 p^{6} T^{18} + 43128 p^{8} T^{20} - 297 p^{10} T^{22} + p^{12} T^{24} \)
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 \( 1 - 15 T + 374 T^{2} - 4485 T^{3} + 70585 T^{4} - 717852 T^{5} + 8690861 T^{6} - 76990419 T^{7} + 777130526 T^{8} - 6072726129 T^{9} + 52865829659 T^{10} - 366568670022 T^{11} + 2805436179305 T^{12} - 366568670022 p T^{13} + 52865829659 p^{2} T^{14} - 6072726129 p^{3} T^{15} + 777130526 p^{4} T^{16} - 76990419 p^{5} T^{17} + 8690861 p^{6} T^{18} - 717852 p^{7} T^{19} + 70585 p^{8} T^{20} - 4485 p^{9} T^{21} + 374 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 8 T - 216 T^{2} - 1192 T^{3} + 31204 T^{4} + 105740 T^{5} - 3167758 T^{6} - 5889708 T^{7} + 250320748 T^{8} + 208010656 T^{9} - 16435908756 T^{10} - 4149850428 T^{11} + 923207251827 T^{12} - 4149850428 p T^{13} - 16435908756 p^{2} T^{14} + 208010656 p^{3} T^{15} + 250320748 p^{4} T^{16} - 5889708 p^{5} T^{17} - 3167758 p^{6} T^{18} + 105740 p^{7} T^{19} + 31204 p^{8} T^{20} - 1192 p^{9} T^{21} - 216 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 - 155 T^{2} + 24317 T^{4} - 2572015 T^{6} + 230267981 T^{8} - 17468522907 T^{10} + 1098167895897 T^{12} - 17468522907 p^{2} T^{14} + 230267981 p^{4} T^{16} - 2572015 p^{6} T^{18} + 24317 p^{8} T^{20} - 155 p^{10} T^{22} + p^{12} T^{24} \)
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 \( 1 + 42 T + 1120 T^{2} + 22344 T^{3} + 367763 T^{4} + 5157765 T^{5} + 63488279 T^{6} + 699015660 T^{7} + 7009629269 T^{8} + 65233277895 T^{9} + 575574698283 T^{10} + 4944431307318 T^{11} + 42178220425467 T^{12} + 4944431307318 p T^{13} + 575574698283 p^{2} T^{14} + 65233277895 p^{3} T^{15} + 7009629269 p^{4} T^{16} + 699015660 p^{5} T^{17} + 63488279 p^{6} T^{18} + 5157765 p^{7} T^{19} + 367763 p^{8} T^{20} + 22344 p^{9} T^{21} + 1120 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 35 T + 407 T^{2} + 1988 T^{3} + 21243 T^{4} + 209523 T^{5} - 2661180 T^{6} - 47561854 T^{7} - 79767523 T^{8} + 341374970 T^{9} - 12037425323 T^{10} + 160472632948 T^{11} + 4058664204571 T^{12} + 160472632948 p T^{13} - 12037425323 p^{2} T^{14} + 341374970 p^{3} T^{15} - 79767523 p^{4} T^{16} - 47561854 p^{5} T^{17} - 2661180 p^{6} T^{18} + 209523 p^{7} T^{19} + 21243 p^{8} T^{20} + 1988 p^{9} T^{21} + 407 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \)
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 \( 1 - 638 T^{2} + 182947 T^{4} - 30717524 T^{6} + 3345408557 T^{8} - 264875421410 T^{10} + 20634099784895 T^{12} - 264875421410 p^{2} T^{14} + 3345408557 p^{4} T^{16} - 30717524 p^{6} T^{18} + 182947 p^{8} T^{20} - 638 p^{10} T^{22} + p^{12} T^{24} \)
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

−3.41547353849683713012757061836, −3.33318677175599731727332108813, −3.27469693279845228844432075586, −3.12144536135279713519948923074, −2.99792996025679915174243461491, −2.94582062405899596132904619327, −2.92912954960312925912149100981, −2.73873889293131526472561419446, −2.52295248866495828648052508882, −2.50866659681620781884978587218, −2.47093829684847117510896947177, −2.05303987804908237864856313611, −1.95163065188968619316251892674, −1.94211240745997086763732667232, −1.93337787714907901298562946484, −1.85803418211040884993124184432, −1.63855227499729993726958633516, −1.62149544095410532181072251436, −1.57563599690564686817966741553, −1.42224929190624269986706684322, −1.28745949726848204392697719818, −1.06586463120164338046030277543, −0.994786452817822426793597153049, −0.793493679917535543387455334819, −0.67118411711015352881383027137, 0.67118411711015352881383027137, 0.793493679917535543387455334819, 0.994786452817822426793597153049, 1.06586463120164338046030277543, 1.28745949726848204392697719818, 1.42224929190624269986706684322, 1.57563599690564686817966741553, 1.62149544095410532181072251436, 1.63855227499729993726958633516, 1.85803418211040884993124184432, 1.93337787714907901298562946484, 1.94211240745997086763732667232, 1.95163065188968619316251892674, 2.05303987804908237864856313611, 2.47093829684847117510896947177, 2.50866659681620781884978587218, 2.52295248866495828648052508882, 2.73873889293131526472561419446, 2.92912954960312925912149100981, 2.94582062405899596132904619327, 2.99792996025679915174243461491, 3.12144536135279713519948923074, 3.27469693279845228844432075586, 3.33318677175599731727332108813, 3.41547353849683713012757061836

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

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