Properties

Label 2-165-1.1-c5-0-0
Degree $2$
Conductor $165$
Sign $1$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08·2-s − 9·3-s − 30.8·4-s + 25·5-s − 9.73·6-s − 139.·7-s − 67.9·8-s + 81·9-s + 27.0·10-s − 121·11-s + 277.·12-s − 646.·13-s − 150.·14-s − 225·15-s + 913.·16-s − 1.37e3·17-s + 87.5·18-s + 1.90e3·19-s − 770.·20-s + 1.25e3·21-s − 130.·22-s + 343.·23-s + 611.·24-s + 625·25-s − 698.·26-s − 729·27-s + 4.30e3·28-s + ⋯
L(s)  = 1  + 0.191·2-s − 0.577·3-s − 0.963·4-s + 0.447·5-s − 0.110·6-s − 1.07·7-s − 0.375·8-s + 0.333·9-s + 0.0854·10-s − 0.301·11-s + 0.556·12-s − 1.06·13-s − 0.205·14-s − 0.258·15-s + 0.891·16-s − 1.15·17-s + 0.0637·18-s + 1.21·19-s − 0.430·20-s + 0.621·21-s − 0.0576·22-s + 0.135·23-s + 0.216·24-s + 0.200·25-s − 0.202·26-s − 0.192·27-s + 1.03·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8809687303\)
\(L(\frac12)\) \(\approx\) \(0.8809687303\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
5 \( 1 - 25T \)
11 \( 1 + 121T \)
good2 \( 1 - 1.08T + 32T^{2} \)
7 \( 1 + 139.T + 1.68e4T^{2} \)
13 \( 1 + 646.T + 3.71e5T^{2} \)
17 \( 1 + 1.37e3T + 1.41e6T^{2} \)
19 \( 1 - 1.90e3T + 2.47e6T^{2} \)
23 \( 1 - 343.T + 6.43e6T^{2} \)
29 \( 1 - 53.5T + 2.05e7T^{2} \)
31 \( 1 - 634.T + 2.86e7T^{2} \)
37 \( 1 - 1.16e4T + 6.93e7T^{2} \)
41 \( 1 - 1.88e4T + 1.15e8T^{2} \)
43 \( 1 - 1.33e4T + 1.47e8T^{2} \)
47 \( 1 + 2.25e3T + 2.29e8T^{2} \)
53 \( 1 + 8.90e3T + 4.18e8T^{2} \)
59 \( 1 + 1.12e4T + 7.14e8T^{2} \)
61 \( 1 - 1.18e4T + 8.44e8T^{2} \)
67 \( 1 + 5.73e4T + 1.35e9T^{2} \)
71 \( 1 - 3.10e4T + 1.80e9T^{2} \)
73 \( 1 - 5.60e4T + 2.07e9T^{2} \)
79 \( 1 + 883.T + 3.07e9T^{2} \)
83 \( 1 - 9.39e4T + 3.93e9T^{2} \)
89 \( 1 - 2.17e4T + 5.58e9T^{2} \)
97 \( 1 + 1.58e5T + 8.58e9T^{2} \)
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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

−12.20927243792349202160507831687, −10.85372426454698831054354432580, −9.638418718604824914349084907357, −9.334602781076864785980735753349, −7.66609064906276933676167276478, −6.39119191606861073606589229219, −5.37990289158597660332594887223, −4.31994535398646703916880005376, −2.77608166455904849518776534731, −0.59035267211140903457833964202, 0.59035267211140903457833964202, 2.77608166455904849518776534731, 4.31994535398646703916880005376, 5.37990289158597660332594887223, 6.39119191606861073606589229219, 7.66609064906276933676167276478, 9.334602781076864785980735753349, 9.638418718604824914349084907357, 10.85372426454698831054354432580, 12.20927243792349202160507831687

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