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Properties

Label 2-597-597.356-c0-0-0
Degree $2$
Conductor $597$
Sign $0.855 - 0.517i$
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  + (0.723 − 0.690i)3-s + (−0.327 + 0.945i)4-s + (−0.419 + 1.72i)7-s + (0.0475 − 0.998i)9-s + (0.415 + 0.909i)12-s + (0.839 + 1.17i)13-s + (−0.786 − 0.618i)16-s + (0.995 − 1.72i)19-s + (0.888 + 1.53i)21-s + (−0.959 − 0.281i)25-s + (−0.654 − 0.755i)27-s + (−1.49 − 0.961i)28-s + (−0.723 − 0.690i)31-s + (0.928 + 0.371i)36-s + (0.142 + 0.246i)37-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)3-s + (−0.327 + 0.945i)4-s + (−0.419 + 1.72i)7-s + (0.0475 − 0.998i)9-s + (0.415 + 0.909i)12-s + (0.839 + 1.17i)13-s + (−0.786 − 0.618i)16-s + (0.995 − 1.72i)19-s + (0.888 + 1.53i)21-s + (−0.959 − 0.281i)25-s + (−0.654 − 0.755i)27-s + (−1.49 − 0.961i)28-s + (−0.723 − 0.690i)31-s + (0.928 + 0.371i)36-s + (0.142 + 0.246i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(597\)    =    \(3 \cdot 199\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(0.297941\)
Root analytic conductor: \(0.545840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{597} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 597,\ (\ :0),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036700796\)
\(L(\frac12)\) \(\approx\) \(1.036700796\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 + 0.690i)T \)
199 \( 1 + (-0.415 - 0.909i)T \)
good2 \( 1 + (0.327 - 0.945i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.419 - 1.72i)T + (-0.888 - 0.458i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.839 - 1.17i)T + (-0.327 + 0.945i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.786 - 0.618i)T^{2} \)
29 \( 1 + (-0.981 - 0.189i)T^{2} \)
31 \( 1 + (0.723 + 0.690i)T + (0.0475 + 0.998i)T^{2} \)
37 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.995 + 0.0950i)T^{2} \)
43 \( 1 + (-0.654 + 1.13i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.888 + 0.458i)T^{2} \)
53 \( 1 + (0.327 + 0.945i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.0671 - 0.466i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.723 - 0.690i)T^{2} \)
73 \( 1 + (0.0311 + 0.0899i)T + (-0.786 + 0.618i)T^{2} \)
79 \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.981 - 0.189i)T^{2} \)
97 \( 1 + (-1.42 - 1.35i)T + (0.0475 + 0.998i)T^{2} \)
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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

−11.52282585526346330785859897496, −9.500289291010045870131806828887, −9.041389031954235447023972924870, −8.555131108404022886689580341542, −7.48038970587739852752493026343, −6.64280438515379600577950778853, −5.60498090575737314949747564736, −4.12564579561348036435225243537, −3.01522905821696337342170251132, −2.19139890288577697202653014789, 1.36844352110050196644713541718, 3.41372956908324390027391074355, 3.99094720263070215736061926094, 5.20351741464072743158793410464, 6.13430420466763690753161245004, 7.49588714867028632850866489426, 8.122487247499142529844969556122, 9.379316758694140791655845353256, 10.01410627951618894512831165953, 10.51326312906908081264906904556

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