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Properties

Label 2-597-597.140-c0-0-0
Degree $2$
Conductor $597$
Sign $-0.848 - 0.529i$
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  + (0.0475 + 0.998i)3-s + (−0.786 + 0.618i)4-s + (−1.03 + 0.531i)7-s + (−0.995 + 0.0950i)9-s + (−0.654 − 0.755i)12-s + (−0.0311 − 0.0899i)13-s + (0.235 − 0.971i)16-s + (−0.981 + 1.70i)19-s + (−0.580 − 1.00i)21-s + (0.841 − 0.540i)25-s + (−0.142 − 0.989i)27-s + (0.481 − 1.05i)28-s + (−0.0475 + 0.998i)31-s + (0.723 − 0.690i)36-s + (0.959 + 1.66i)37-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)3-s + (−0.786 + 0.618i)4-s + (−1.03 + 0.531i)7-s + (−0.995 + 0.0950i)9-s + (−0.654 − 0.755i)12-s + (−0.0311 − 0.0899i)13-s + (0.235 − 0.971i)16-s + (−0.981 + 1.70i)19-s + (−0.580 − 1.00i)21-s + (0.841 − 0.540i)25-s + (−0.142 − 0.989i)27-s + (0.481 − 1.05i)28-s + (−0.0475 + 0.998i)31-s + (0.723 − 0.690i)36-s + (0.959 + 1.66i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ -0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5561464174\)
\(L(\frac12)\) \(\approx\) \(0.5561464174\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0475 - 0.998i)T \)
199 \( 1 + (0.654 + 0.755i)T \)
good2 \( 1 + (0.786 - 0.618i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (1.03 - 0.531i)T + (0.580 - 0.814i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.0311 + 0.0899i)T + (-0.786 + 0.618i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.235 - 0.971i)T^{2} \)
29 \( 1 + (-0.928 + 0.371i)T^{2} \)
31 \( 1 + (0.0475 - 0.998i)T + (-0.995 - 0.0950i)T^{2} \)
37 \( 1 + (-0.959 - 1.66i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (-0.142 + 0.246i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.580 + 0.814i)T^{2} \)
53 \( 1 + (0.786 + 0.618i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.0475 + 0.998i)T^{2} \)
73 \( 1 + (-1.56 - 1.23i)T + (0.235 + 0.971i)T^{2} \)
79 \( 1 + (0.308 - 1.27i)T + (-0.888 - 0.458i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.928 + 0.371i)T^{2} \)
97 \( 1 + (-0.0883 + 1.85i)T + (-0.995 - 0.0950i)T^{2} \)
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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

−11.11186962879733996825084090702, −10.03252016321846211614363011892, −9.658504640187885914556742999390, −8.592233573685446117708749516044, −8.194161610810371177292894681489, −6.61311329777548918790385428608, −5.64268443587608118489214060329, −4.58883919826198307581413555704, −3.66383731559038897271070015378, −2.81861273458891305495316654939, 0.65100470936188839641489486218, 2.43238993349445116730358191131, 3.83154509181851250020901350840, 5.04741733678553910911655077783, 6.19114733149397442679026630219, 6.79583011642622299492307770378, 7.80712733914045979560074502660, 8.997359735217611463660736888785, 9.384773119894284644460302426241, 10.60577910140164690921918090531

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