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Properties

Label 4-45e2-1.1-c7e2-0-3
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 197.608197.608
Root an. cond. 3.749313.74931
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·2-s − 69·4-s − 250·5-s + 1.30e3·7-s + 413·8-s + 1.75e3·10-s − 3.44e3·11-s − 8.98e3·13-s − 9.12e3·14-s − 4.86e3·16-s + 5.49e3·17-s − 4.95e4·19-s + 1.72e4·20-s + 2.41e4·22-s − 9.18e4·23-s + 4.68e4·25-s + 6.29e4·26-s − 8.99e4·28-s − 1.81e5·29-s + 3.04e5·31-s + 1.61e5·32-s − 3.84e4·34-s − 3.26e5·35-s − 5.02e5·37-s + 3.47e5·38-s − 1.03e5·40-s − 6.31e5·41-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.539·4-s − 0.894·5-s + 1.43·7-s + 0.285·8-s + 0.553·10-s − 0.781·11-s − 1.13·13-s − 0.889·14-s − 0.296·16-s + 0.271·17-s − 1.65·19-s + 0.482·20-s + 0.483·22-s − 1.57·23-s + 3/5·25-s + 0.702·26-s − 0.774·28-s − 1.38·29-s + 1.83·31-s + 0.872·32-s − 0.167·34-s − 1.28·35-s − 1.63·37-s + 1.02·38-s − 0.255·40-s − 1.43·41-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 197.608197.608
Root analytic conductor: 3.749313.74931
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2025, ( :7/2,7/2), 1)(4,\ 2025,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1+p3T)2 ( 1 + p^{3} T )^{2}
good2D4D_{4} 1+7T+59pT2+7p7T3+p14T4 1 + 7 T + 59 p T^{2} + 7 p^{7} T^{3} + p^{14} T^{4}
7D4D_{4} 1−1304T+1601006T2−1304p7T3+p14T4 1 - 1304 T + 1601006 T^{2} - 1304 p^{7} T^{3} + p^{14} T^{4}
11D4D_{4} 1+3448T+9598294T2+3448p7T3+p14T4 1 + 3448 T + 9598294 T^{2} + 3448 p^{7} T^{3} + p^{14} T^{4}
13D4D_{4} 1+8988T+43676926T2+8988p7T3+p14T4 1 + 8988 T + 43676926 T^{2} + 8988 p^{7} T^{3} + p^{14} T^{4}
17D4D_{4} 1−5492T+533727862T2−5492p7T3+p14T4 1 - 5492 T + 533727862 T^{2} - 5492 p^{7} T^{3} + p^{14} T^{4}
19D4D_{4} 1+49584T+1767259558T2+49584p7T3+p14T4 1 + 49584 T + 1767259558 T^{2} + 49584 p^{7} T^{3} + p^{14} T^{4}
23D4D_{4} 1+91848T+4843394254T2+91848p7T3+p14T4 1 + 91848 T + 4843394254 T^{2} + 91848 p^{7} T^{3} + p^{14} T^{4}
29D4D_{4} 1+6268pT+32439203278T2+6268p8T3+p14T4 1 + 6268 p T + 32439203278 T^{2} + 6268 p^{8} T^{3} + p^{14} T^{4}
31D4D_{4} 1−304232T+77458297022T2−304232p7T3+p14T4 1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4}
37D4D_{4} 1+502316T+221684315886T2+502316p7T3+p14T4 1 + 502316 T + 221684315886 T^{2} + 502316 p^{7} T^{3} + p^{14} T^{4}
41D4D_{4} 1+631172T+420346017142T2+631172p7T3+p14T4 1 + 631172 T + 420346017142 T^{2} + 631172 p^{7} T^{3} + p^{14} T^{4}
43D4D_{4} 1−353640T+567251429590T2−353640p7T3+p14T4 1 - 353640 T + 567251429590 T^{2} - 353640 p^{7} T^{3} + p^{14} T^{4}
47D4D_{4} 1−467480T+1062629128990T2−467480p7T3+p14T4 1 - 467480 T + 1062629128990 T^{2} - 467480 p^{7} T^{3} + p^{14} T^{4}
53D4D_{4} 1−568052T+2403786403294T2−568052p7T3+p14T4 1 - 568052 T + 2403786403294 T^{2} - 568052 p^{7} T^{3} + p^{14} T^{4}
59D4D_{4} 1+287224T+1627391637238T2+287224p7T3+p14T4 1 + 287224 T + 1627391637238 T^{2} + 287224 p^{7} T^{3} + p^{14} T^{4}
61D4D_{4} 1+2514180T+7865442419758T2+2514180p7T3+p14T4 1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4}
67D4D_{4} 1+5073832T+16021265240102T2+5073832p7T3+p14T4 1 + 5073832 T + 16021265240102 T^{2} + 5073832 p^{7} T^{3} + p^{14} T^{4}
71D4D_{4} 1−3748816T+20824804809646T2−3748816p7T3+p14T4 1 - 3748816 T + 20824804809646 T^{2} - 3748816 p^{7} T^{3} + p^{14} T^{4}
73D4D_{4} 1+1477212T−3158190986T2+1477212p7T3+p14T4 1 + 1477212 T - 3158190986 T^{2} + 1477212 p^{7} T^{3} + p^{14} T^{4}
79D4D_{4} 1+4627720T+42789559383518T2+4627720p7T3+p14T4 1 + 4627720 T + 42789559383518 T^{2} + 4627720 p^{7} T^{3} + p^{14} T^{4}
83D4D_{4} 1−6072936T+62224060120582T2−6072936p7T3+p14T4 1 - 6072936 T + 62224060120582 T^{2} - 6072936 p^{7} T^{3} + p^{14} T^{4}
89D4D_{4} 1+16516356T+156597055746838T2+16516356p7T3+p14T4 1 + 16516356 T + 156597055746838 T^{2} + 16516356 p^{7} T^{3} + p^{14} T^{4}
97D4D_{4} 1−2723428T+135845063471622T2−2723428p7T3+p14T4 1 - 2723428 T + 135845063471622 T^{2} - 2723428 p^{7} T^{3} + p^{14} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.84188907858769980231030012621, −13.81849805991726470654828166429, −12.64154293168540013826529166892, −12.22829697892357241591768583173, −11.64710674077183346086845416618, −11.03745075735779380247780328528, −10.27130159313228053604390998013, −9.939056423156428691375938383141, −8.734492645305948054844981074791, −8.554147740585570774472803405260, −7.81092460717480919964266114578, −7.47085272984639799998415172378, −6.34083011669719464819863907502, −5.20345335328960321441633260041, −4.61207387956342506577495486947, −4.00798575400841684145448791088, −2.57140154513953339983378026514, −1.62901188374473814017497986832, 0, 0, 1.62901188374473814017497986832, 2.57140154513953339983378026514, 4.00798575400841684145448791088, 4.61207387956342506577495486947, 5.20345335328960321441633260041, 6.34083011669719464819863907502, 7.47085272984639799998415172378, 7.81092460717480919964266114578, 8.554147740585570774472803405260, 8.734492645305948054844981074791, 9.939056423156428691375938383141, 10.27130159313228053604390998013, 11.03745075735779380247780328528, 11.64710674077183346086845416618, 12.22829697892357241591768583173, 12.64154293168540013826529166892, 13.81849805991726470654828166429, 13.84188907858769980231030012621

Graph of the ZZ-function along the critical line