Properties

Label 2-286650-1.1-c1-0-135
Degree $2$
Conductor $286650$
Sign $1$
Analytic cond. $2288.91$
Root an. cond. $47.8425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(2288.91\)
Root analytic conductor: \(47.8425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 286650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.869835776\)
\(L(\frac12)\) \(\approx\) \(4.869835776\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line