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Properties

Label 4-975e2-1.1-c0e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 2·13-s − 16-s + 4·27-s − 4·39-s − 2·48-s + 2·49-s + 5·81-s − 6·117-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·3-s + 3·9-s − 2·13-s − 16-s + 4·27-s − 4·39-s − 2·48-s + 2·49-s + 5·81-s − 6·117-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-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 1.8483778001.848377800
L(12)L(\frac12) ≈\approx 1.8483778001.848377800
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)
bad3C1C_1 (1−T)2 ( 1 - T )^{2}
5 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1+T4 1 + T^{4}
7C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
11C22C_2^2 1+T4 1 + T^{4}
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{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}
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C22C_2^2 1+T4 1 + T^{4}
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C22C_2^2 1+T4 1 + T^{4}
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C22C_2^2 1+T4 1 + T^{4}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
71C22C_2^2 1+T4 1 + T^{4}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
83C22C_2^2 1+T4 1 + T^{4}
89C22C_2^2 1+T4 1 + T^{4}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
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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

−10.17934329951944876245294953206, −9.973438318531000944450321310781, −9.445482340411059171270096194668, −9.083492567683835646843068218470, −8.915114406647258837726466396188, −8.407566945802334014804710733946, −7.82001538497289228716159363266, −7.58543290177525607492175639340, −7.24588872077008628092812186312, −6.80616561924932108697696819786, −6.44021501339372778440570961299, −5.57818974090550141849119573935, −4.84824727925279207794426501916, −4.77124347468273750850239592743, −3.98677422463913272559462774347, −3.77565726747821271093854660763, −2.90525958001038370072008469250, −2.43833327113116172909390124467, −2.30666693957681143269879376228, −1.37193100007792083659560202861, 1.37193100007792083659560202861, 2.30666693957681143269879376228, 2.43833327113116172909390124467, 2.90525958001038370072008469250, 3.77565726747821271093854660763, 3.98677422463913272559462774347, 4.77124347468273750850239592743, 4.84824727925279207794426501916, 5.57818974090550141849119573935, 6.44021501339372778440570961299, 6.80616561924932108697696819786, 7.24588872077008628092812186312, 7.58543290177525607492175639340, 7.82001538497289228716159363266, 8.407566945802334014804710733946, 8.915114406647258837726466396188, 9.083492567683835646843068218470, 9.445482340411059171270096194668, 9.973438318531000944450321310781, 10.17934329951944876245294953206

Graph of the ZZ-function along the critical line