Properties

Label 2-270802-1.1-c1-0-3
Degree $2$
Conductor $270802$
Sign $1$
Analytic cond. $2162.36$
Root an. cond. $46.5012$
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 + 2·5-s − 7-s − 8-s − 3·9-s − 2·10-s − 4·11-s + 6·13-s + 14-s + 16-s − 6·17-s + 3·18-s − 4·19-s + 2·20-s + 4·22-s + 23-s − 25-s − 6·26-s − 28-s + 4·31-s − 32-s + 6·34-s − 2·35-s − 3·36-s − 6·37-s + 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270802\)    =    \(2 \cdot 7 \cdot 23 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(2162.36\)
Root analytic conductor: \(46.5012\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{270802} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8869751118\)
\(L(\frac12)\) \(\approx\) \(0.8869751118\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−12.86452900267107, −12.37823800754816, −11.66594817988425, −11.21330175805049, −10.80290936042007, −10.56875647197070, −10.09087296534446, −9.415921072813983, −9.013043807733617, −8.697742929209510, −8.189184680632136, −7.907471425173676, −7.033704474153313, −6.564743956539690, −6.219939579953747, −5.728448075013940, −5.421556536696974, −4.617518279189994, −4.044684334330194, −3.321858538201944, −2.809747321103142, −2.238540705273906, −1.935252090229171, −1.038463676302126, −0.3031935288831658, 0.3031935288831658, 1.038463676302126, 1.935252090229171, 2.238540705273906, 2.809747321103142, 3.321858538201944, 4.044684334330194, 4.617518279189994, 5.421556536696974, 5.728448075013940, 6.219939579953747, 6.564743956539690, 7.033704474153313, 7.907471425173676, 8.189184680632136, 8.697742929209510, 9.013043807733617, 9.415921072813983, 10.09087296534446, 10.56875647197070, 10.80290936042007, 11.21330175805049, 11.66594817988425, 12.37823800754816, 12.86452900267107

Graph of the $Z$-function along the critical line