Properties

Degree $2$
Conductor $182$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s − 3·6-s + 7-s − 8-s + 6·9-s − 5·11-s + 3·12-s − 13-s − 14-s + 16-s − 4·17-s − 6·18-s + 2·19-s + 3·21-s + 5·22-s + 5·23-s − 3·24-s − 5·25-s + 26-s + 9·27-s + 28-s + 4·29-s + 31-s − 32-s − 15·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 1.50·11-s + 0.866·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.41·18-s + 0.458·19-s + 0.654·21-s + 1.06·22-s + 1.04·23-s − 0.612·24-s − 25-s + 0.196·26-s + 1.73·27-s + 0.188·28-s + 0.742·29-s + 0.179·31-s − 0.176·32-s − 2.61·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{182} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.393197700\)
\(L(\frac12)\) \(\approx\) \(1.393197700\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−19.91872357419125, −18.85357701601020, −18.34053007444472, −17.37090009301296, −15.97800408457365, −15.39268410548274, −14.66511588983303, −13.53334905123550, −13.00106062133414, −11.48257101074862, −10.27042276265821, −9.528699368682305, −8.410332621243878, −7.959201586617155, −6.905165873746227, −4.887388377509635, −3.164647028724546, −2.106814980024164, 2.106814980024164, 3.164647028724546, 4.887388377509635, 6.905165873746227, 7.959201586617155, 8.410332621243878, 9.528699368682305, 10.27042276265821, 11.48257101074862, 13.00106062133414, 13.53334905123550, 14.66511588983303, 15.39268410548274, 15.97800408457365, 17.37090009301296, 18.34053007444472, 18.85357701601020, 19.91872357419125

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