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Properties

Label 4-31360-1.1-c1e2-0-9
Degree 44
Conductor 3136031360
Sign −1-1
Analytic cond. 1.999541.99954
Root an. cond. 1.189131.18913
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s − 3·7-s + 8-s − 5·9-s − 2·10-s − 11-s − 3·13-s − 3·14-s + 16-s − 5·18-s − 2·20-s − 22-s − 2·25-s − 3·26-s − 3·28-s + 31-s + 32-s + 6·35-s − 5·36-s − 2·40-s − 2·43-s − 44-s + 10·45-s − 6·47-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s − 5/3·9-s − 0.632·10-s − 0.301·11-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 1.17·18-s − 0.447·20-s − 0.213·22-s − 2/5·25-s − 0.588·26-s − 0.566·28-s + 0.179·31-s + 0.176·32-s + 1.01·35-s − 5/6·36-s − 0.316·40-s − 0.304·43-s − 0.150·44-s + 1.49·45-s − 0.875·47-s + 2/7·49-s + ⋯

Functional equation

Λ(s)=(31360s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(31360s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3136031360    =    27⋅5⋅722^{7} \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 1.999541.99954
Root analytic conductor: 1.189131.18913
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 31360, ( :1/2,1/2), −1)(4,\ 31360,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2C1C_1 1−T 1 - T
5C1C_1×\timesC2C_2 (1+T)(1+T+pT2) ( 1 + T )( 1 + T + p T^{2} )
7C2C_2 1+3T+pT2 1 + 3 T + p T^{2}
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2×\timesC2C_2 (1−2T+pT2)(1+3T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2×\timesC2C_2 (1−T+pT2)(1+4T+pT2) ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
19C22C_2^2 1−12T2+p2T4 1 - 12 T^{2} + p^{2} T^{4}
23C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
29C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−8T+pT2)(1+7T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
41C22C_2^2 1+43T2+p2T4 1 + 43 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−2T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C22C_2^2 1+60T2+p2T4 1 + 60 T^{2} + p^{2} T^{4}
59C22C_2^2 1−52T2+p2T4 1 - 52 T^{2} + p^{2} T^{4}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2×\timesC2C_2 (1−3T+pT2)(1+7T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} )
71C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
73C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
79C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
83C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
89C22C_2^2 1+143T2+p2T4 1 + 143 T^{2} + p^{2} T^{4}
97C22C_2^2 1−120T2+p2T4 1 - 120 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

−10.30625158281814654646136601008, −9.662116873776763503704987997104, −9.290253827775280498435333499559, −8.389606765016871261693289332634, −8.163601736289340551621444198358, −7.49141551790521137416786289005, −6.81577031659550740336496387543, −6.38032383064270632520363812271, −5.60956005351140354594733538219, −5.25056001774719032595567424552, −4.35698877117903131179237458107, −3.60640018363373500720888002371, −3.04296613229166822081277949951, −2.40461908035731444788935104109, 0, 2.40461908035731444788935104109, 3.04296613229166822081277949951, 3.60640018363373500720888002371, 4.35698877117903131179237458107, 5.25056001774719032595567424552, 5.60956005351140354594733538219, 6.38032383064270632520363812271, 6.81577031659550740336496387543, 7.49141551790521137416786289005, 8.163601736289340551621444198358, 8.389606765016871261693289332634, 9.290253827775280498435333499559, 9.662116873776763503704987997104, 10.30625158281814654646136601008

Graph of the ZZ-function along the critical line