Properties

Label 2-405-1.1-c1-0-14
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  + 0.732·2-s − 1.46·4-s + 5-s − 4.73·7-s − 2.53·8-s + 0.732·10-s − 5.73·11-s + 1.46·13-s − 3.46·14-s + 1.07·16-s − 2.73·17-s + 4.46·19-s − 1.46·20-s − 4.19·22-s − 3.46·23-s + 25-s + 1.07·26-s + 6.92·28-s + 3.19·29-s − 3·31-s + 5.85·32-s − 2·34-s − 4.73·35-s − 2.73·37-s + 3.26·38-s − 2.53·40-s − 7.19·41-s + ⋯
L(s)  = 1  + 0.517·2-s − 0.732·4-s + 0.447·5-s − 1.78·7-s − 0.896·8-s + 0.231·10-s − 1.72·11-s + 0.406·13-s − 0.925·14-s + 0.267·16-s − 0.662·17-s + 1.02·19-s − 0.327·20-s − 0.894·22-s − 0.722·23-s + 0.200·25-s + 0.210·26-s + 1.30·28-s + 0.593·29-s − 0.538·31-s + 1.03·32-s − 0.342·34-s − 0.799·35-s − 0.449·37-s + 0.530·38-s − 0.400·40-s − 1.12·41-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 \( 1 - 0.732T + 2T^{2} \)
7 \( 1 + 4.73T + 7T^{2} \)
11 \( 1 + 5.73T + 11T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 2.73T + 17T^{2} \)
19 \( 1 - 4.46T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 - 0.196T + 43T^{2} \)
47 \( 1 + 8.73T + 47T^{2} \)
53 \( 1 - 6.73T + 53T^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + 7.66T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + 9.66T + 97T^{2} \)
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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

−10.53395455354729580116053469386, −9.891659987097994186587409226244, −9.160920829589346233820277538329, −8.113279120425039489990623479181, −6.77562535273111037724955155460, −5.84110997200682682606964101848, −5.05003835867806828695676183738, −3.61864026751503349114017566624, −2.75549156342552231674726398882, 0, 2.75549156342552231674726398882, 3.61864026751503349114017566624, 5.05003835867806828695676183738, 5.84110997200682682606964101848, 6.77562535273111037724955155460, 8.113279120425039489990623479181, 9.160920829589346233820277538329, 9.891659987097994186587409226244, 10.53395455354729580116053469386

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