Properties

Label 4-810e2-1.1-c3e2-0-1
Degree 44
Conductor 656100656100
Sign 11
Analytic cond. 2284.032284.03
Root an. cond. 6.913146.91314
Motivic weight 33
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  − 2·2-s + 5·5-s + 4·7-s + 8·8-s − 10·10-s + 48·11-s − 2·13-s − 8·14-s − 16·16-s − 228·17-s + 280·19-s − 96·22-s − 72·23-s + 4·26-s − 210·29-s − 272·31-s + 456·34-s + 20·35-s − 668·37-s − 560·38-s + 40·40-s + 198·41-s + 268·43-s + 144·46-s − 216·47-s + 343·49-s − 156·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s + 0.215·7-s + 0.353·8-s − 0.316·10-s + 1.31·11-s − 0.0426·13-s − 0.152·14-s − 1/4·16-s − 3.25·17-s + 3.38·19-s − 0.930·22-s − 0.652·23-s + 0.0301·26-s − 1.34·29-s − 1.57·31-s + 2.30·34-s + 0.0965·35-s − 2.96·37-s − 2.39·38-s + 0.158·40-s + 0.754·41-s + 0.950·43-s + 0.461·46-s − 0.670·47-s + 49-s − 0.404·53-s + ⋯

Functional equation

Λ(s)=(656100s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(656100s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 656100656100    =    2238522^{2} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 2284.032284.03
Root analytic conductor: 6.913146.91314
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 656100, ( :3/2,3/2), 1)(4,\ 656100,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 0.48685362880.4868536288
L(12)L(\frac12) \approx 0.48685362880.4868536288
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+pT+p2T2 1 + p T + p^{2} T^{2}
3 1 1
5C2C_2 1pT+p2T2 1 - p T + p^{2} T^{2}
good7C22C_2^2 14T327T24p3T3+p6T4 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 148T+973T248p3T3+p6T4 1 - 48 T + 973 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4}
13C22C_2^2 1+2T2193T2+2p3T3+p6T4 1 + 2 T - 2193 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1+114T+p3T2)2 ( 1 + 114 T + p^{3} T^{2} )^{2}
19C2C_2 (1140T+p3T2)2 ( 1 - 140 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+72T6983T2+72p3T3+p6T4 1 + 72 T - 6983 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
29C22C_2^2 1+210T+19711T2+210p3T3+p6T4 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4}
31C22C_2^2 1+272T+44193T2+272p3T3+p6T4 1 + 272 T + 44193 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4}
37C2C_2 (1+334T+p3T2)2 ( 1 + 334 T + p^{3} T^{2} )^{2}
41C22C_2^2 1198T29717T2198p3T3+p6T4 1 - 198 T - 29717 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1268T7683T2268p3T3+p6T4 1 - 268 T - 7683 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4}
47C22C_2^2 1+216T57167T2+216p3T3+p6T4 1 + 216 T - 57167 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4}
53C2C_2 (1+78T+p3T2)2 ( 1 + 78 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+240T147779T2+240p3T3+p6T4 1 + 240 T - 147779 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1+302T135777T2+302p3T3+p6T4 1 + 302 T - 135777 T^{2} + 302 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1+596T+54453T2+596p3T3+p6T4 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1+768T+p3T2)2 ( 1 + 768 T + p^{3} T^{2} )^{2}
73C2C_2 (1+478T+p3T2)2 ( 1 + 478 T + p^{3} T^{2} )^{2}
79C22C_2^2 1640T83439T2640p3T3+p6T4 1 - 640 T - 83439 T^{2} - 640 p^{3} T^{3} + p^{6} T^{4}
83C22C_2^2 1348T450683T2348p3T3+p6T4 1 - 348 T - 450683 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4}
89C2C_2 (1210T+p3T2)2 ( 1 - 210 T + p^{3} T^{2} )^{2}
97C22C_2^2 11534T+1440483T21534p3T3+p6T4 1 - 1534 T + 1440483 T^{2} - 1534 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=p j=14(1αj,pps)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

−9.948803142664595719246401922463, −9.294011631793388166509074229030, −9.250988954048808336708468966002, −8.984159280521740295314032792267, −8.737245772324103545045651916639, −7.81110752509252653158896263146, −7.53731043249234003750357538343, −7.04557397410445080895579418777, −6.85223204447930698287103087803, −6.20675495835752627143755411642, −5.52004068151849288517290720147, −5.42876452954204143358306610172, −4.59802095882482836037123259004, −4.21461360619803631157018387916, −3.63328139636967747284214443138, −3.09250217730309355128884354598, −2.23787650472173326400753020356, −1.59847307293441694674770582672, −1.39266072303511398267761898096, −0.20970422872946751674271682152, 0.20970422872946751674271682152, 1.39266072303511398267761898096, 1.59847307293441694674770582672, 2.23787650472173326400753020356, 3.09250217730309355128884354598, 3.63328139636967747284214443138, 4.21461360619803631157018387916, 4.59802095882482836037123259004, 5.42876452954204143358306610172, 5.52004068151849288517290720147, 6.20675495835752627143755411642, 6.85223204447930698287103087803, 7.04557397410445080895579418777, 7.53731043249234003750357538343, 7.81110752509252653158896263146, 8.737245772324103545045651916639, 8.984159280521740295314032792267, 9.250988954048808336708468966002, 9.294011631793388166509074229030, 9.948803142664595719246401922463

Graph of the ZZ-function along the critical line