Properties

Label 2-161-1.1-c1-0-2
Degree $2$
Conductor $161$
Sign $1$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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 + 3·8-s − 3·9-s − 2·10-s + 4·11-s + 6·13-s − 14-s − 16-s − 2·17-s + 3·18-s + 4·19-s − 2·20-s − 4·22-s − 23-s − 25-s − 6·26-s − 28-s − 2·29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s − 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-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 − 0.371·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $1$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8628343267\)
\(L(\frac12)\) \(\approx\) \(0.8628343267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.12082677402822691324891147054, −11.57306814136850016603459846709, −10.82999474885334929942270674109, −9.519763597988292185167511678467, −8.955336427994798867614764820369, −8.048209611049889306918613094768, −6.43402806030236505610147737889, −5.38200398568328423768601134305, −3.76496444511642110376256706933, −1.51926854854030366908096170303, 1.51926854854030366908096170303, 3.76496444511642110376256706933, 5.38200398568328423768601134305, 6.43402806030236505610147737889, 8.048209611049889306918613094768, 8.955336427994798867614764820369, 9.519763597988292185167511678467, 10.82999474885334929942270674109, 11.57306814136850016603459846709, 13.12082677402822691324891147054

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