Properties

Label 2-585-1.1-c1-0-6
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + 0.289·2-s − 1.91·4-s + 5-s + 4.91·7-s − 1.13·8-s + 0.289·10-s − 4.91·11-s + 13-s + 1.42·14-s + 3.50·16-s + 4.33·17-s + 2.57·19-s − 1.91·20-s − 1.42·22-s + 6.33·23-s + 25-s + 0.289·26-s − 9.42·28-s − 6·29-s + 1.42·31-s + 3.27·32-s + 1.25·34-s + 4.91·35-s + 9.49·37-s + 0.745·38-s − 1.13·40-s − 4.33·41-s + ⋯
L(s)  = 1  + 0.204·2-s − 0.958·4-s + 0.447·5-s + 1.85·7-s − 0.400·8-s + 0.0914·10-s − 1.48·11-s + 0.277·13-s + 0.379·14-s + 0.876·16-s + 1.05·17-s + 0.591·19-s − 0.428·20-s − 0.303·22-s + 1.32·23-s + 0.200·25-s + 0.0567·26-s − 1.78·28-s − 1.11·29-s + 0.255·31-s + 0.579·32-s + 0.215·34-s + 0.831·35-s + 1.56·37-s + 0.120·38-s − 0.179·40-s − 0.677·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.655208056\)
\(L(\frac12)\) \(\approx\) \(1.655208056\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 - 0.289T + 2T^{2} \)
7 \( 1 - 4.91T + 7T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
17 \( 1 - 4.33T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 - 6.33T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 - 9.49T + 37T^{2} \)
41 \( 1 + 4.33T + 41T^{2} \)
43 \( 1 + 1.15T + 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 - 0.338T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 7.25T + 67T^{2} \)
71 \( 1 + 0.916T + 71T^{2} \)
73 \( 1 + 3.15T + 73T^{2} \)
79 \( 1 + 3.49T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 - 0.338T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−10.71006556545615916011925360722, −9.853626577935644136413324711716, −8.855626152911665993410041039110, −8.046068407393169179994503054766, −7.47087379872489673523363566142, −5.54332531963769101314586777057, −5.28994245545142214570415822163, −4.32044664385178774185364284532, −2.84739340004738343312459938622, −1.25490646077372766887393526375, 1.25490646077372766887393526375, 2.84739340004738343312459938622, 4.32044664385178774185364284532, 5.28994245545142214570415822163, 5.54332531963769101314586777057, 7.47087379872489673523363566142, 8.046068407393169179994503054766, 8.855626152911665993410041039110, 9.853626577935644136413324711716, 10.71006556545615916011925360722

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