Properties

Label 2-405-1.1-c1-0-9
Degree $2$
Conductor $405$
Sign $-1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·2-s + 5.46·4-s + 5-s − 1.26·7-s − 9.46·8-s − 2.73·10-s − 2.26·11-s − 5.46·13-s + 3.46·14-s + 14.9·16-s + 0.732·17-s − 2.46·19-s + 5.46·20-s + 6.19·22-s + 3.46·23-s + 25-s + 14.9·26-s − 6.92·28-s − 7.19·29-s − 3·31-s − 21.8·32-s − 2·34-s − 1.26·35-s + 0.732·37-s + 6.73·38-s − 9.46·40-s + 3.19·41-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.73·4-s + 0.447·5-s − 0.479·7-s − 3.34·8-s − 0.863·10-s − 0.683·11-s − 1.51·13-s + 0.925·14-s + 3.73·16-s + 0.177·17-s − 0.565·19-s + 1.22·20-s + 1.32·22-s + 0.722·23-s + 0.200·25-s + 2.92·26-s − 1.30·28-s − 1.33·29-s − 0.538·31-s − 3.86·32-s − 0.342·34-s − 0.214·35-s + 0.120·37-s + 1.09·38-s − 1.49·40-s + 0.499·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
good2 \( 1 + 2.73T + 2T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 - 0.732T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 7.19T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 0.732T + 37T^{2} \)
41 \( 1 - 3.19T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 5.26T + 47T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + 0.267T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 - 8.53T + 79T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 7.66T + 97T^{2} \)
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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

−10.47901817533946089705980451277, −9.722832552934395336803449497777, −9.222697266578735014282348495178, −8.103368373131588726861552509454, −7.31996317957000879743322008349, −6.50951253662799041173106328615, −5.30600954979009837211133848160, −2.99718508586738244117959337162, −1.94056688123354518976348832487, 0, 1.94056688123354518976348832487, 2.99718508586738244117959337162, 5.30600954979009837211133848160, 6.50951253662799041173106328615, 7.31996317957000879743322008349, 8.103368373131588726861552509454, 9.222697266578735014282348495178, 9.722832552934395336803449497777, 10.47901817533946089705980451277

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