Properties

Label 2-405-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.571·2-s − 1.67·4-s − 5-s + 1.42·7-s + 2.10·8-s + 0.571·10-s − 2.67·11-s + 4.67·13-s − 0.816·14-s + 2.14·16-s − 2.67·17-s + 4.67·19-s + 1.67·20-s + 1.52·22-s + 5.91·23-s + 25-s − 2.67·26-s − 2.38·28-s + 9.48·29-s + 6.96·31-s − 5.42·32-s + 1.52·34-s − 1.42·35-s − 1.81·37-s − 2.67·38-s − 2.10·40-s + 1.47·41-s + ⋯
L(s)  = 1  − 0.404·2-s − 0.836·4-s − 0.447·5-s + 0.539·7-s + 0.742·8-s + 0.180·10-s − 0.805·11-s + 1.29·13-s − 0.218·14-s + 0.535·16-s − 0.648·17-s + 1.07·19-s + 0.374·20-s + 0.325·22-s + 1.23·23-s + 0.200·25-s − 0.524·26-s − 0.451·28-s + 1.76·29-s + 1.25·31-s − 0.959·32-s + 0.262·34-s − 0.241·35-s − 0.298·37-s − 0.433·38-s − 0.332·40-s + 0.229·41-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: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9252771142\)
\(L(\frac12)\) \(\approx\) \(0.9252771142\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.571T + 2T^{2} \)
7 \( 1 - 1.42T + 7T^{2} \)
11 \( 1 + 2.67T + 11T^{2} \)
13 \( 1 - 4.67T + 13T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 - 5.91T + 23T^{2} \)
29 \( 1 - 9.48T + 29T^{2} \)
31 \( 1 - 6.96T + 31T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 - 0.471T + 43T^{2} \)
47 \( 1 + 6.95T + 47T^{2} \)
53 \( 1 - 1.14T + 53T^{2} \)
59 \( 1 - 1.14T + 59T^{2} \)
61 \( 1 + 2.52T + 61T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + 0.287T + 79T^{2} \)
83 \( 1 - 4.28T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 7.83T + 97T^{2} \)
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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

−11.07038247212309512576822396450, −10.40031686953871304086596644506, −9.306546041442950052020454251449, −8.397483001033902170637563682929, −7.939104289954117485209762841929, −6.66347051677547516233021240332, −5.22666800839247342637974200460, −4.46965434298240444207065051840, −3.13793794920203995567181359917, −1.05768489668298083823781522462, 1.05768489668298083823781522462, 3.13793794920203995567181359917, 4.46965434298240444207065051840, 5.22666800839247342637974200460, 6.66347051677547516233021240332, 7.939104289954117485209762841929, 8.397483001033902170637563682929, 9.306546041442950052020454251449, 10.40031686953871304086596644506, 11.07038247212309512576822396450

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