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Properties

Label 2-597-597.239-c0-0-0
Degree $2$
Conductor $597$
Sign $0.267 - 0.963i$
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  + (0.580 + 0.814i)3-s + (0.723 + 0.690i)4-s + (−1.95 − 0.186i)7-s + (−0.327 + 0.945i)9-s + (−0.142 + 0.989i)12-s + (1.07 + 0.431i)13-s + (0.0475 + 0.998i)16-s + (0.786 − 1.36i)19-s + (−0.981 − 1.70i)21-s + (0.415 − 0.909i)25-s + (−0.959 + 0.281i)27-s + (−1.28 − 1.48i)28-s + (−0.580 + 0.814i)31-s + (−0.888 + 0.458i)36-s + (−0.841 − 1.45i)37-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (0.723 + 0.690i)4-s + (−1.95 − 0.186i)7-s + (−0.327 + 0.945i)9-s + (−0.142 + 0.989i)12-s + (1.07 + 0.431i)13-s + (0.0475 + 0.998i)16-s + (0.786 − 1.36i)19-s + (−0.981 − 1.70i)21-s + (0.415 − 0.909i)25-s + (−0.959 + 0.281i)27-s + (−1.28 − 1.48i)28-s + (−0.580 + 0.814i)31-s + (−0.888 + 0.458i)36-s + (−0.841 − 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.267 - 0.963i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.267 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083711738\)
\(L(\frac12)\) \(\approx\) \(1.083711738\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 - 0.814i)T \)
199 \( 1 + (0.142 - 0.989i)T \)
good2 \( 1 + (-0.723 - 0.690i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (1.95 + 0.186i)T + (0.981 + 0.189i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-1.07 - 0.431i)T + (0.723 + 0.690i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0475 + 0.998i)T^{2} \)
29 \( 1 + (-0.235 - 0.971i)T^{2} \)
31 \( 1 + (0.580 - 0.814i)T + (-0.327 - 0.945i)T^{2} \)
37 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.786 + 0.618i)T^{2} \)
43 \( 1 + (-0.959 + 1.66i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.981 - 0.189i)T^{2} \)
53 \( 1 + (-0.723 + 0.690i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (0.473 - 0.451i)T + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (0.0135 + 0.284i)T + (-0.995 + 0.0950i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.235 - 0.971i)T^{2} \)
97 \( 1 + (-0.273 + 0.384i)T + (-0.327 - 0.945i)T^{2} \)
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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

−10.81167542806653673497350573401, −10.29909582599192748389082534783, −9.059236407292905569427153494774, −8.858289404649049722535607732322, −7.36678439989778498996191046751, −6.75074342874153452682899141014, −5.69968088828314023806964633201, −4.06458562339766127366896559850, −3.36973695479247728876350481451, −2.55811005660805821472361694245, 1.37264522470789985047242948895, 2.91676258302836788227610288923, 3.51401255389757596696970877161, 5.76333424608857868578009829957, 6.18386108319548503447106449390, 7.00537379183000026673207856005, 7.906262040992545785104668433418, 9.140867544190095059379825137356, 9.724871832779496381897700653116, 10.58898008017526878176798441273

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