Properties

Label 2-405-1.1-c1-0-8
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + 2·2-s + 2·4-s + 5-s + 2·10-s + 5·11-s + 4·13-s − 4·16-s − 4·17-s − 5·19-s + 2·20-s + 10·22-s + 6·23-s + 25-s + 8·26-s − 5·29-s − 9·31-s − 8·32-s − 8·34-s − 10·37-s − 10·38-s + 7·41-s − 2·43-s + 10·44-s + 12·46-s + 2·47-s − 7·49-s + 2·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 1.50·11-s + 1.10·13-s − 16-s − 0.970·17-s − 1.14·19-s + 0.447·20-s + 2.13·22-s + 1.25·23-s + 1/5·25-s + 1.56·26-s − 0.928·29-s − 1.61·31-s − 1.41·32-s − 1.37·34-s − 1.64·37-s − 1.62·38-s + 1.09·41-s − 0.304·43-s + 1.50·44-s + 1.76·46-s + 0.291·47-s − 49-s + 0.282·50-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: yes
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.957866634\)
\(L(\frac12)\) \(\approx\) \(2.957866634\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−11.31548780266267451963425405875, −10.84103814713817700544872039181, −9.161742231918082706521428553592, −8.824991276786141939896068673100, −6.94257636940465251066789252595, −6.35526976205125359438346080603, −5.41659698665533312416876873683, −4.23466268327432431147155666300, −3.49410387761142976656546809271, −1.89799498723585767421033230965, 1.89799498723585767421033230965, 3.49410387761142976656546809271, 4.23466268327432431147155666300, 5.41659698665533312416876873683, 6.35526976205125359438346080603, 6.94257636940465251066789252595, 8.824991276786141939896068673100, 9.161742231918082706521428553592, 10.84103814713817700544872039181, 11.31548780266267451963425405875

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