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Properties

Label 4-950e2-1.1-c1e2-0-12
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 57.544157.5441
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 6-s + 8·7-s + 8-s + 3·9-s + 6·11-s + 2·13-s − 8·14-s − 16-s − 6·17-s − 3·18-s − 7·19-s + 8·21-s − 6·22-s − 6·23-s + 24-s − 2·26-s + 8·27-s + 4·31-s + 6·33-s + 6·34-s + 20·37-s + 7·38-s + 2·39-s − 9·41-s − 8·42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s + 3.02·7-s + 0.353·8-s + 9-s + 1.80·11-s + 0.554·13-s − 2.13·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.60·19-s + 1.74·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s − 0.392·26-s + 1.53·27-s + 0.718·31-s + 1.04·33-s + 1.02·34-s + 3.28·37-s + 1.13·38-s + 0.320·39-s − 1.40·41-s − 1.23·42-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 57.544157.5441
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :1/2,1/2), 1)(4,\ 902500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1895887743.189588774
L(12)L(\frac12) ≈\approx 3.1895887743.189588774
L(32)L(\frac{3}{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)
bad2C2C_2 1+T+T2 1 + T + T^{2}
5 1 1
19C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good3C22C_2^2 1−T−2T2−pT3+p2T4 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−7T+pT2)(1+5T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} )
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+6T+13T2+6pT3+p2T4 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C22C_2^2 1+9T+40T2+9pT3+p2T4 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+4T−27T2+4pT3+p2T4 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−9T+22T2−9pT3+p2T4 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−4T−45T2−4pT3+p2T4 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+7T−18T2+7pT3+p2T4 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−6T−35T2−6pT3+p2T4 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+T−72T2+pT3+p2T4 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4}
79C2C_2 (1−17T+pT2)(1+13T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−17T+192T2−17pT3+p2T4 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} 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

−10.09474199711844011687907176169, −9.946850356494535096082866129794, −9.149570185205668476990639362047, −8.874323509406739257606047355231, −8.519249016691948735719943177521, −8.345774465162069181527750648356, −7.81236668075384802346719820142, −7.69992835747480889570578386681, −6.72485337033489874047842306545, −6.68664673243537788932100393543, −6.15593470953801047883776310921, −5.36774904382666829579776527488, −4.65934366186294047675663120380, −4.33593392557876322772710668377, −4.30568778485884231597679763653, −3.73474008897104265396212364368, −2.30369861584046849330064066884, −2.19508762073766577431119049007, −1.39598196030327509443736632745, −1.12913565088584557117681843960, 1.12913565088584557117681843960, 1.39598196030327509443736632745, 2.19508762073766577431119049007, 2.30369861584046849330064066884, 3.73474008897104265396212364368, 4.30568778485884231597679763653, 4.33593392557876322772710668377, 4.65934366186294047675663120380, 5.36774904382666829579776527488, 6.15593470953801047883776310921, 6.68664673243537788932100393543, 6.72485337033489874047842306545, 7.69992835747480889570578386681, 7.81236668075384802346719820142, 8.345774465162069181527750648356, 8.519249016691948735719943177521, 8.874323509406739257606047355231, 9.149570185205668476990639362047, 9.946850356494535096082866129794, 10.09474199711844011687907176169

Graph of the ZZ-function along the critical line