↑

Properties

Label 4-975e2-1.1-c0e2-0-4
Degree 44
Conductor 950625950625
Sign 11
Analytic cond. 0.2367680.236768
Root an. cond. 0.6975580.697558
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 9-s + 3·16-s − 2·36-s − 2·49-s − 4·61-s + 4·64-s + 4·79-s + 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + ⋯
L(s)  = 1  + 2·4-s − 9-s + 3·16-s − 2·36-s − 2·49-s − 4·61-s + 4·64-s + 4·79-s + 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + ⋯

Functional equation

Λ(s)=(950625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(950625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 950625950625    =    32⋅54⋅1323^{2} \cdot 5^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.2367680.236768
Root analytic conductor: 0.6975580.697558
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 950625, ( :0,0), 1)(4,\ 950625,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4901318921.490131892
L(12)L(\frac12) ≈\approx 1.4901318921.490131892
L(1)L(1) 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)
bad3C2C_2 1+T2 1 + T^{2}
5 1 1
13C2C_2 1+T2 1 + T^{2}
good2C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
11C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
19C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C1C_1 (1+T)4 ( 1 + T )^{4}
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
79C1C_1 (1−T)4 ( 1 - T )^{4}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
show more
show less
   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

−10.46020180672746986598322320198, −10.25459188500847120213976126290, −9.600924897993804394861135573876, −9.145070378795872887496922649413, −8.841380063270587955574540273338, −7.937506276045764260070505889456, −7.87464050505085899264970070378, −7.71768273400817361388107516167, −6.87859619786778195954961749199, −6.59237856878386894559391487066, −6.08149711375738883685380744832, −6.07656575759035098726143849334, −5.13072896459162593903227548049, −5.06959454151953004867126607954, −4.07246558130023193839670554749, −3.35201431561297263149031426010, −3.11187708242688814260298900724, −2.53427834112161155281039054535, −1.96214344693111957840067107755, −1.32628486048628880277635760772, 1.32628486048628880277635760772, 1.96214344693111957840067107755, 2.53427834112161155281039054535, 3.11187708242688814260298900724, 3.35201431561297263149031426010, 4.07246558130023193839670554749, 5.06959454151953004867126607954, 5.13072896459162593903227548049, 6.07656575759035098726143849334, 6.08149711375738883685380744832, 6.59237856878386894559391487066, 6.87859619786778195954961749199, 7.71768273400817361388107516167, 7.87464050505085899264970070378, 7.937506276045764260070505889456, 8.841380063270587955574540273338, 9.145070378795872887496922649413, 9.600924897993804394861135573876, 10.25459188500847120213976126290, 10.46020180672746986598322320198

Graph of the ZZ-function along the critical line