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Properties

Label 2-597-597.224-c0-0-0
Degree $2$
Conductor $597$
Sign $0.0152 + 0.999i$
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  + (−0.327 − 0.945i)3-s + (0.0475 − 0.998i)4-s + (1.82 − 0.351i)7-s + (−0.786 + 0.618i)9-s + (−0.959 + 0.281i)12-s + (−0.473 + 0.451i)13-s + (−0.995 − 0.0950i)16-s + (−0.235 + 0.408i)19-s + (−0.928 − 1.60i)21-s + (−0.654 + 0.755i)25-s + (0.841 + 0.540i)27-s + (−0.264 − 1.83i)28-s + (0.327 − 0.945i)31-s + (0.580 + 0.814i)36-s + (−0.415 − 0.719i)37-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)3-s + (0.0475 − 0.998i)4-s + (1.82 − 0.351i)7-s + (−0.786 + 0.618i)9-s + (−0.959 + 0.281i)12-s + (−0.473 + 0.451i)13-s + (−0.995 − 0.0950i)16-s + (−0.235 + 0.408i)19-s + (−0.928 − 1.60i)21-s + (−0.654 + 0.755i)25-s + (0.841 + 0.540i)27-s + (−0.264 − 1.83i)28-s + (0.327 − 0.945i)31-s + (0.580 + 0.814i)36-s + (−0.415 − 0.719i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0152 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0152 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.0152 + 0.999i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.0152 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9271020223\)
\(L(\frac12)\) \(\approx\) \(0.9271020223\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
199 \( 1 + (0.959 - 0.281i)T \)
good2 \( 1 + (-0.0475 + 0.998i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-1.82 + 0.351i)T + (0.928 - 0.371i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.473 - 0.451i)T + (0.0475 - 0.998i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.995 - 0.0950i)T^{2} \)
29 \( 1 + (0.888 + 0.458i)T^{2} \)
31 \( 1 + (-0.327 + 0.945i)T + (-0.786 - 0.618i)T^{2} \)
37 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.235 + 0.971i)T^{2} \)
43 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.928 + 0.371i)T^{2} \)
53 \( 1 + (-0.0475 - 0.998i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.327 - 0.945i)T^{2} \)
73 \( 1 + (0.0748 + 1.57i)T + (-0.995 + 0.0950i)T^{2} \)
79 \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.888 + 0.458i)T^{2} \)
97 \( 1 + (-0.581 + 1.67i)T + (-0.786 - 0.618i)T^{2} \)
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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

−10.94573082388081168502135571364, −9.956340009936685061737328101991, −8.781853400810014459175129536957, −7.84567166802998870071420700912, −7.18195384000603191849786976124, −6.05678162432974886491834794388, −5.24160784721145529091562962402, −4.40460896122563917894330073381, −2.17387941774602575814779108182, −1.35974585115395871837604838665, 2.27907012506855842024326058175, 3.59501766870448941308369567339, 4.71123984558014212547693585785, 5.18847562296229606673726971955, 6.62415420704875082885444786075, 7.913196291439653677414818312467, 8.370867809527857556960676847212, 9.239186155760816644357763860798, 10.44160985971282373605048902008, 11.10894205004148217104418444345

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