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Properties

Label 4-950e2-1.1-c1e2-0-11
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 8·7-s + 8-s + 3·9-s − 2·11-s − 2·13-s − 8·14-s − 16-s + 3·17-s − 3·18-s + 8·19-s + 2·22-s − 4·23-s + 2·26-s + 6·29-s − 4·31-s − 3·34-s − 4·37-s − 8·38-s + 3·41-s + 12·43-s + 4·46-s + 6·47-s + 34·49-s + 4·53-s + 8·56-s − 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 3.02·7-s + 0.353·8-s + 9-s − 0.603·11-s − 0.554·13-s − 2.13·14-s − 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s + 0.426·22-s − 0.834·23-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.514·34-s − 0.657·37-s − 1.29·38-s + 0.468·41-s + 1.82·43-s + 0.589·46-s + 0.875·47-s + 34/7·49-s + 0.549·53-s + 1.06·56-s − 0.787·58-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 2.5082892872.508289287
L(12)L(\frac12) ≈\approx 2.5082892872.508289287
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 1−8T+pT2 1 - 8 T + p T^{2}
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
13C2C_2 (1−5T+pT2)(1+7T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1−3T−8T2−3pT3+p2T4 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+4T−7T2+4pT3+p2T4 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−6T+7T2−6pT3+p2T4 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1−3T−32T2−3pT3+p2T4 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−12T+101T2−12pT3+p2T4 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−6T−11T2−6pT3+p2T4 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−4T−37T2−4pT3+p2T4 1 - 4 T - 37 T^{2} - 4 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 1−12T+83T2−12pT3+p2T4 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−15T+158T2−15pT3+p2T4 1 - 15 T + 158 T^{2} - 15 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}
73C2C_2 (1−17T+pT2)(1+7T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} )
79C22C_2^2 1−14T+117T2−14pT3+p2T4 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4}
83C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
89C22C_2^2 1−T−88T2−pT3+p2T4 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4}
97C22C_2^2 1+T−96T2+pT3+p2T4 1 + T - 96 T^{2} + 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

−10.38450045593392763058529638412, −9.806945868444657179890078391459, −9.331278319024421238205381058932, −9.131648018292945956548558624444, −8.304064873109822080349928709952, −8.160092209846945261948423177638, −7.74449496139154088767916504511, −7.59955754260840875186267303582, −7.16683350197101172488231284634, −6.64712956023782217981691171034, −5.58134872982081990061053675284, −5.37910592099521989355386862677, −5.13721948123108281567837603195, −4.48585311121576836448253852871, −4.21259459093084646413143670001, −3.55849083310874862777676318559, −2.47815361197550296732342454157, −2.13962698116482825132393866715, −1.21933920243924547826322587540, −1.09921874520690091350756836785, 1.09921874520690091350756836785, 1.21933920243924547826322587540, 2.13962698116482825132393866715, 2.47815361197550296732342454157, 3.55849083310874862777676318559, 4.21259459093084646413143670001, 4.48585311121576836448253852871, 5.13721948123108281567837603195, 5.37910592099521989355386862677, 5.58134872982081990061053675284, 6.64712956023782217981691171034, 7.16683350197101172488231284634, 7.59955754260840875186267303582, 7.74449496139154088767916504511, 8.160092209846945261948423177638, 8.304064873109822080349928709952, 9.131648018292945956548558624444, 9.331278319024421238205381058932, 9.806945868444657179890078391459, 10.38450045593392763058529638412

Graph of the ZZ-function along the critical line