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Properties

Label 2-592-148.31-c1-0-5
Degree $2$
Conductor $592$
Sign $0.763 - 0.646i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.64·3-s + (−1 + i)5-s − 2.64i·7-s + 4.00·9-s − 2.64·11-s + (−1 + i)13-s + (2.64 − 2.64i)15-s + (2 − 2i)17-s + 7.00i·21-s + (2.64 + 2.64i)23-s + 3i·25-s − 2.64·27-s + (7 + 7i)29-s + 7.00·33-s + (2.64 + 2.64i)35-s + ⋯
L(s)  = 1  − 1.52·3-s + (−0.447 + 0.447i)5-s − 0.999i·7-s + 1.33·9-s − 0.797·11-s + (−0.277 + 0.277i)13-s + (0.683 − 0.683i)15-s + (0.485 − 0.485i)17-s + 1.52i·21-s + (0.551 + 0.551i)23-s + 0.600i·25-s − 0.509·27-s + (1.29 + 1.29i)29-s + 1.21·33-s + (0.447 + 0.447i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.763 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600498 + 0.220160i\)
\(L(\frac12)\) \(\approx\) \(0.600498 + 0.220160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-1 + 6i)T \)
good3 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-2 + 2i)T - 17iT^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + (-2.64 - 2.64i)T + 23iT^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 - 31iT^{2} \)
41 \( 1 - 3iT - 41T^{2} \)
43 \( 1 + (-7.93 - 7.93i)T + 43iT^{2} \)
47 \( 1 - 7.93iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-5.29 - 5.29i)T + 59iT^{2} \)
61 \( 1 + (6 + 6i)T + 61iT^{2} \)
67 \( 1 + 5.29T + 67T^{2} \)
71 \( 1 - 2.64iT - 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 + (-5.29 - 5.29i)T + 79iT^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (-4 - 4i)T + 89iT^{2} \)
97 \( 1 + (-6 + 6i)T - 97iT^{2} \)
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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

−10.79647454982643989393612904684, −10.36735765566232031661927404125, −9.257896559051280656395406034196, −7.65744431100030758845054161817, −7.22408801751207204610309097889, −6.26169152924263748092345048178, −5.21615931314666885300694326619, −4.45823645112189294637637164889, −3.11597850888458860760440926805, −0.959187043391705670538466054835, 0.59510735106584512424673630890, 2.56104864384797019233987783835, 4.31131575419316987530905883747, 5.22275929305860832064320969445, 5.81569157495462038847668696402, 6.76774666521787070377219255644, 7.979843858296724315537815128746, 8.726665904173579082644473728943, 10.06494778022201382938784444349, 10.59305676990730435575837863332

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