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Properties

Label 4-6776e2-1.1-c1e2-0-9
Degree 44
Conductor 4591417645914176
Sign 11
Analytic cond. 2927.522927.52
Root an. cond. 7.355727.35572
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s + 2·7-s − 9-s + 2·13-s + 3·15-s + 4·17-s + 2·19-s − 2·21-s − 7·23-s + 25-s + 4·29-s − 17·31-s − 6·35-s + 5·37-s − 2·39-s + 20·41-s − 8·43-s + 3·45-s + 4·47-s + 3·49-s − 4·51-s + 8·53-s − 2·57-s − 15·59-s − 8·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 0.755·7-s − 1/3·9-s + 0.554·13-s + 0.774·15-s + 0.970·17-s + 0.458·19-s − 0.436·21-s − 1.45·23-s + 1/5·25-s + 0.742·29-s − 3.05·31-s − 1.01·35-s + 0.821·37-s − 0.320·39-s + 3.12·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s + 3/7·49-s − 0.560·51-s + 1.09·53-s − 0.264·57-s − 1.95·59-s − 1.02·61-s − 0.251·63-s + ⋯

Functional equation

Λ(s)=(45914176s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45914176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(45914176s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45914176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4591417645914176    =    26⋅72⋅1142^{6} \cdot 7^{2} \cdot 11^{4}
Sign: 11
Analytic conductor: 2927.522927.52
Root analytic conductor: 7.355727.35572
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 45914176, ( :1/2,1/2), 1)(4,\ 45914176,\ (\ :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)
bad2 1 1
7C1C_1 (1−T)2 ( 1 - T )^{2}
11 1 1
good3D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19D4D_{4} 1−2T+22T2−2pT3+p2T4 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+7T+54T2+7pT3+p2T4 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31D4D_{4} 1+17T+130T2+17pT3+p2T4 1 + 17 T + 130 T^{2} + 17 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−5T+76T2−5pT3+p2T4 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+15T+170T2+15pT3+p2T4 1 + 15 T + 170 T^{2} + 15 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+8T+70T2+8pT3+p2T4 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+T+130T2+pT3+p2T4 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4}
71D4D_{4} 1+5T+110T2+5pT3+p2T4 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+8T+94T2+8pT3+p2T4 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+2T+142T2+2pT3+p2T4 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4}
83C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
89D4D_{4} 1+21T+284T2+21pT3+p2T4 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−13T+232T2−13pT3+p2T4 1 - 13 T + 232 T^{2} - 13 p T^{3} + 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

−7.77267234439599828279592930901, −7.52807244477523977203161271699, −7.29786509197751487805626380862, −6.87206484717902914504319962563, −6.19944182302415655194744581296, −5.99982328875424387942571806626, −5.57960540307878158835134197716, −5.54285930231328525424556792476, −4.98929567408074377397827291788, −4.30888889305250728453149327499, −4.24815577570749729812860521751, −3.96871247875509430894966132480, −3.32702265487708601549182623848, −3.22644033963672142558675996812, −2.45615586923298913303358862890, −2.07656987191789601156764518747, −1.30942469510242811071717685274, −1.11032867739664032807415032881, 0, 0, 1.11032867739664032807415032881, 1.30942469510242811071717685274, 2.07656987191789601156764518747, 2.45615586923298913303358862890, 3.22644033963672142558675996812, 3.32702265487708601549182623848, 3.96871247875509430894966132480, 4.24815577570749729812860521751, 4.30888889305250728453149327499, 4.98929567408074377397827291788, 5.54285930231328525424556792476, 5.57960540307878158835134197716, 5.99982328875424387942571806626, 6.19944182302415655194744581296, 6.87206484717902914504319962563, 7.29786509197751487805626380862, 7.52807244477523977203161271699, 7.77267234439599828279592930901

Graph of the ZZ-function along the critical line