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Properties

Label 2-597-597.386-c0-0-0
Degree $2$
Conductor $597$
Sign $0.724 + 0.689i$
Analytic cond. $0.297941$
Root an. cond. $0.545840$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.235 − 0.971i)3-s + (0.981 + 0.189i)4-s + (0.0688 − 0.0656i)7-s + (−0.888 − 0.458i)9-s + (0.415 − 0.909i)12-s + (−0.469 − 0.0448i)13-s + (0.928 + 0.371i)16-s + (−0.580 + 1.00i)19-s + (−0.0475 − 0.0824i)21-s + (−0.959 + 0.281i)25-s + (−0.654 + 0.755i)27-s + (0.0800 − 0.0514i)28-s + (−0.235 − 0.971i)31-s + (−0.786 − 0.618i)36-s + (0.142 + 0.246i)37-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)3-s + (0.981 + 0.189i)4-s + (0.0688 − 0.0656i)7-s + (−0.888 − 0.458i)9-s + (0.415 − 0.909i)12-s + (−0.469 − 0.0448i)13-s + (0.928 + 0.371i)16-s + (−0.580 + 1.00i)19-s + (−0.0475 − 0.0824i)21-s + (−0.959 + 0.281i)25-s + (−0.654 + 0.755i)27-s + (0.0800 − 0.0514i)28-s + (−0.235 − 0.971i)31-s + (−0.786 − 0.618i)36-s + (0.142 + 0.246i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.724 + 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143583456\)
\(L(\frac12)\) \(\approx\) \(1.143583456\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 + 0.971i)T \)
199 \( 1 + (-0.415 + 0.909i)T \)
good2 \( 1 + (-0.981 - 0.189i)T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.0688 + 0.0656i)T + (0.0475 - 0.998i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.469 + 0.0448i)T + (0.981 + 0.189i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.928 + 0.371i)T^{2} \)
29 \( 1 + (0.327 - 0.945i)T^{2} \)
31 \( 1 + (0.235 + 0.971i)T + (-0.888 + 0.458i)T^{2} \)
37 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.580 - 0.814i)T^{2} \)
43 \( 1 + (-0.654 + 1.13i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.0475 + 0.998i)T^{2} \)
53 \( 1 + (-0.981 + 0.189i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.264 - 1.83i)T + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.235 - 0.971i)T^{2} \)
73 \( 1 + (1.74 - 0.336i)T + (0.928 - 0.371i)T^{2} \)
79 \( 1 + (-0.771 - 0.308i)T + (0.723 + 0.690i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (0.327 - 0.945i)T^{2} \)
97 \( 1 + (0.154 + 0.635i)T + (-0.888 + 0.458i)T^{2} \)
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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

−10.94591349414884248693898971833, −10.00127407501776418482706637306, −8.836567605641451484864253663812, −7.79602089673729426439920282608, −7.40810519452475856486780213599, −6.32340779369991160856651625583, −5.67661050679805087770067790502, −3.90577789441203536638758558951, −2.67212464545260518670272895703, −1.70111839046561994341950038939, 2.17477596042661381556723970633, 3.18374050132810608496073802211, 4.45043553775640887361666625428, 5.44824364415725494496992757976, 6.43443663239105089307060268405, 7.46421924264639042093653768100, 8.425992964438066853719095455881, 9.410855049422205770268204270315, 10.17908367543101548063691404617, 10.94840138581397051351829660916

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