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Properties

Label 4-950e2-1.1-c1e2-0-2
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  − 4-s − 3·9-s − 8·11-s + 16-s − 2·19-s + 6·29-s − 4·31-s + 3·36-s − 12·41-s + 8·44-s − 11·49-s + 18·59-s − 24·61-s − 64-s + 2·76-s + 4·79-s − 4·89-s + 24·99-s − 16·101-s − 38·109-s − 6·116-s + 26·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 9-s − 2.41·11-s + 1/4·16-s − 0.458·19-s + 1.11·29-s − 0.718·31-s + 1/2·36-s − 1.87·41-s + 1.20·44-s − 1.57·49-s + 2.34·59-s − 3.07·61-s − 1/8·64-s + 0.229·76-s + 0.450·79-s − 0.423·89-s + 2.41·99-s − 1.59·101-s − 3.63·109-s − 0.557·116-s + 2.36·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 0.32701774960.3270177496
L(12)L(\frac12) ≈\approx 0.32701774960.3270177496
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+T2 1 + T^{2}
5 1 1
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
23C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1+63T2+p2T4 1 + 63 T^{2} + p^{2} T^{4}
59C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
61C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
79C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
83C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 T^{2} + 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

−10.15016545704303514162933428331, −10.13005754909113889890701048623, −9.244549850815312309721075645447, −9.093704631429224472444185127512, −8.340239925267233839920725068411, −8.267657670479892378188304515591, −7.900926595129397868586337442323, −7.48816219465705187636854610825, −6.64126552377188817121822222429, −6.59378533420009187135711703750, −5.72012460110250655705893751189, −5.33685527201188894107005962850, −5.21063747931156207296261021631, −4.59185789638246659387353467167, −4.06614574231571688955308805184, −3.17293555578459101267045758585, −2.98335211476737464017735058856, −2.39703959808082716387771161589, −1.60717875927706914596224029356, −0.25849936757337310997818913150, 0.25849936757337310997818913150, 1.60717875927706914596224029356, 2.39703959808082716387771161589, 2.98335211476737464017735058856, 3.17293555578459101267045758585, 4.06614574231571688955308805184, 4.59185789638246659387353467167, 5.21063747931156207296261021631, 5.33685527201188894107005962850, 5.72012460110250655705893751189, 6.59378533420009187135711703750, 6.64126552377188817121822222429, 7.48816219465705187636854610825, 7.900926595129397868586337442323, 8.267657670479892378188304515591, 8.340239925267233839920725068411, 9.093704631429224472444185127512, 9.244549850815312309721075645447, 10.13005754909113889890701048623, 10.15016545704303514162933428331

Graph of the ZZ-function along the critical line