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Properties

Label 4-975e2-1.1-c0e2-0-1
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  + 3-s − 4-s − 7-s − 12-s − 2·13-s + 19-s − 21-s − 27-s + 28-s + 4·31-s + 2·37-s − 2·39-s − 43-s + 49-s + 2·52-s + 57-s − 2·61-s + 64-s − 67-s + 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s + 4·93-s + 2·97-s + ⋯
L(s)  = 1  + 3-s − 4-s − 7-s − 12-s − 2·13-s + 19-s − 21-s − 27-s + 28-s + 4·31-s + 2·37-s − 2·39-s − 43-s + 49-s + 2·52-s + 57-s − 2·61-s + 64-s − 67-s + 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s + 4·93-s + 2·97-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 0.77957867640.7795786764
L(12)L(\frac12) ≈\approx 0.77957867640.7795786764
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−T+T2 1 - T + T^{2}
5 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good2C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
7C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C1C_1 (1−T)4 ( 1 - T )^{4}
37C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
67C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
71C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
73C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
79C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
97C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{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

−10.14481120059794087714542727938, −9.852246864281772762571094234387, −9.610824892434926778972806506377, −9.184768914145084891096575728697, −8.926594045490684485566430927972, −8.223345515667233378018295425003, −8.143038880934048142478579951979, −7.41785223496469855228093816686, −7.37145874042251345594491716672, −6.54631896126174916189396117275, −6.20060069090052131309275776188, −5.73813529112427156566665246733, −4.92930580491230472922330850948, −4.57442467153360372299369917498, −4.46368752189094048442954809538, −3.51420551010126304392432655651, −3.04054851144198347176414363508, −2.70408263627851848195748200723, −2.19207251486768035259803088450, −0.813275479504411650719544841679, 0.813275479504411650719544841679, 2.19207251486768035259803088450, 2.70408263627851848195748200723, 3.04054851144198347176414363508, 3.51420551010126304392432655651, 4.46368752189094048442954809538, 4.57442467153360372299369917498, 4.92930580491230472922330850948, 5.73813529112427156566665246733, 6.20060069090052131309275776188, 6.54631896126174916189396117275, 7.37145874042251345594491716672, 7.41785223496469855228093816686, 8.143038880934048142478579951979, 8.223345515667233378018295425003, 8.926594045490684485566430927972, 9.184768914145084891096575728697, 9.610824892434926778972806506377, 9.852246864281772762571094234387, 10.14481120059794087714542727938

Graph of the ZZ-function along the critical line