Properties

Label 2-165-1.1-c5-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.25·2-s + 9·3-s + 53.6·4-s + 25·5-s − 83.2·6-s + 36.4·7-s − 200.·8-s + 81·9-s − 231.·10-s − 121·11-s + 482.·12-s − 878.·13-s − 337.·14-s + 225·15-s + 138.·16-s + 155.·17-s − 749.·18-s − 1.93e3·19-s + 1.34e3·20-s + 328.·21-s + 1.11e3·22-s + 1.92e3·23-s − 1.80e3·24-s + 625·25-s + 8.12e3·26-s + 729·27-s + 1.95e3·28-s + ⋯
L(s)  = 1  − 1.63·2-s + 0.577·3-s + 1.67·4-s + 0.447·5-s − 0.944·6-s + 0.281·7-s − 1.10·8-s + 0.333·9-s − 0.731·10-s − 0.301·11-s + 0.968·12-s − 1.44·13-s − 0.459·14-s + 0.258·15-s + 0.135·16-s + 0.130·17-s − 0.545·18-s − 1.22·19-s + 0.749·20-s + 0.162·21-s + 0.493·22-s + 0.759·23-s − 0.639·24-s + 0.200·25-s + 2.35·26-s + 0.192·27-s + 0.471·28-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: \(1\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.25T + 32T^{2} \)
7 \( 1 - 36.4T + 1.68e4T^{2} \)
13 \( 1 + 878.T + 3.71e5T^{2} \)
17 \( 1 - 155.T + 1.41e6T^{2} \)
19 \( 1 + 1.93e3T + 2.47e6T^{2} \)
23 \( 1 - 1.92e3T + 6.43e6T^{2} \)
29 \( 1 + 480.T + 2.05e7T^{2} \)
31 \( 1 - 1.75e3T + 2.86e7T^{2} \)
37 \( 1 + 1.89e3T + 6.93e7T^{2} \)
41 \( 1 + 4.50e3T + 1.15e8T^{2} \)
43 \( 1 + 4.47e3T + 1.47e8T^{2} \)
47 \( 1 + 1.23e4T + 2.29e8T^{2} \)
53 \( 1 - 2.14e3T + 4.18e8T^{2} \)
59 \( 1 + 1.58e4T + 7.14e8T^{2} \)
61 \( 1 + 3.64e4T + 8.44e8T^{2} \)
67 \( 1 + 1.56e4T + 1.35e9T^{2} \)
71 \( 1 - 1.06e4T + 1.80e9T^{2} \)
73 \( 1 - 1.21e4T + 2.07e9T^{2} \)
79 \( 1 + 8.72e4T + 3.07e9T^{2} \)
83 \( 1 - 9.72e4T + 3.93e9T^{2} \)
89 \( 1 - 3.86e4T + 5.58e9T^{2} \)
97 \( 1 + 3.67e4T + 8.58e9T^{2} \)
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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

−10.95234006513408669213722949422, −10.11555758995383618390600633519, −9.369676587058756600020413038646, −8.446637262097494723882953486691, −7.58765644839194753908363378024, −6.60430373285099731449965007460, −4.81813386704793197314484513086, −2.69442915745644190984397759167, −1.65207071322074254971014894369, 0, 1.65207071322074254971014894369, 2.69442915745644190984397759167, 4.81813386704793197314484513086, 6.60430373285099731449965007460, 7.58765644839194753908363378024, 8.446637262097494723882953486691, 9.369676587058756600020413038646, 10.11555758995383618390600633519, 10.95234006513408669213722949422

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