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Properties

Label 2-286650-1.1-c1-0-135
Degree 22
Conductor 286650286650
Sign 11
Analytic cond. 2288.912288.91
Root an. cond. 47.842547.8425
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 4·11-s − 13-s + 16-s + 6·17-s − 8·19-s + 4·22-s + 2·23-s − 26-s − 2·29-s + 32-s + 6·34-s − 2·37-s − 8·38-s + 6·41-s + 4·44-s + 2·46-s − 8·47-s − 52-s − 12·53-s − 2·58-s − 4·59-s + 10·61-s + 64-s − 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s − 0.277·13-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.852·22-s + 0.417·23-s − 0.196·26-s − 0.371·29-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 1.29·38-s + 0.937·41-s + 0.603·44-s + 0.294·46-s − 1.16·47-s − 0.138·52-s − 1.64·53-s − 0.262·58-s − 0.520·59-s + 1.28·61-s + 1/8·64-s − 0.488·67-s + ⋯

Functional equation

Λ(s)=(286650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(286650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 286650286650    =    2⋅32⋅52⋅72⋅132 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 2288.912288.91
Root analytic conductor: 47.842547.8425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 286650, ( :1/2), 1)(2,\ 286650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.8698357764.869835776
L(12)L(\frac12) ≈\approx 4.8698357764.869835776
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}
ppFp(T)F_p(T)
bad2 1−T 1 - T
3 1 1
5 1 1
7 1 1
13 1+T 1 + T
good11 1−4T+pT2 1 - 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1−12T+pT2 1 - 12 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.69986281514039, −12.27768872828186, −11.98149384435848, −11.43648762556289, −10.84786952917001, −10.68979634108553, −9.965337526200196, −9.465214630206532, −9.192137088221824, −8.441865744416544, −8.042292749428528, −7.596966395589063, −6.888065040526156, −6.600476758973026, −6.080241273024196, −5.711353942327281, −4.914251456681605, −4.685652610043172, −4.005847886180981, −3.521162641669052, −3.195060123084524, −2.312666488674694, −1.893118659940161, −1.234313371826712, −0.5288834619075988, 0.5288834619075988, 1.234313371826712, 1.893118659940161, 2.312666488674694, 3.195060123084524, 3.521162641669052, 4.005847886180981, 4.685652610043172, 4.914251456681605, 5.711353942327281, 6.080241273024196, 6.600476758973026, 6.888065040526156, 7.596966395589063, 8.042292749428528, 8.441865744416544, 9.192137088221824, 9.465214630206532, 9.965337526200196, 10.68979634108553, 10.84786952917001, 11.43648762556289, 11.98149384435848, 12.27768872828186, 12.69986281514039

Graph of the ZZ-function along the critical line