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Properties

Label 2-592-148.103-c1-0-0
Degree $2$
Conductor $592$
Sign $-0.713 + 0.701i$
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  + (−1.36 + 2.36i)3-s + (0.702 + 2.62i)5-s + (−2.97 − 1.71i)7-s + (−2.23 − 3.86i)9-s − 0.327·11-s + (1.36 + 5.09i)13-s + (−7.15 − 1.91i)15-s + (−2.97 − 0.795i)17-s + (−3.12 + 0.836i)19-s + (8.12 − 4.68i)21-s + (−0.136 − 0.136i)23-s + (−2.04 + 1.17i)25-s + 4.00·27-s + (−5.38 − 5.38i)29-s + (5.98 − 5.98i)31-s + ⋯
L(s)  = 1  + (−0.788 + 1.36i)3-s + (0.314 + 1.17i)5-s + (−1.12 − 0.648i)7-s + (−0.744 − 1.28i)9-s − 0.0988·11-s + (0.378 + 1.41i)13-s + (−1.84 − 0.495i)15-s + (−0.720 − 0.193i)17-s + (−0.715 + 0.191i)19-s + (1.77 − 1.02i)21-s + (−0.0284 − 0.0284i)23-s + (−0.408 + 0.235i)25-s + 0.769·27-s + (−0.999 − 0.999i)29-s + (1.07 − 1.07i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.713 + 0.701i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.713 + 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162143 - 0.396223i\)
\(L(\frac12)\) \(\approx\) \(0.162143 - 0.396223i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 5.80i)T \)
good3 \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.702 - 2.62i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.97 + 1.71i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.327T + 11T^{2} \)
13 \( 1 + (-1.36 - 5.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.97 + 0.795i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.12 - 0.836i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.136 + 0.136i)T + 23iT^{2} \)
29 \( 1 + (5.38 + 5.38i)T + 29iT^{2} \)
31 \( 1 + (-5.98 + 5.98i)T - 31iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.88 - 4.88i)T + 43iT^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + (5.43 + 9.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.55 - 9.52i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.67 - 2.59i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.16 - 5.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.5 + 6.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.81iT - 73T^{2} \)
79 \( 1 + (-0.649 + 0.174i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (9.32 - 5.38i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.63 - 6.11i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.0 - 12.0i)T - 97iT^{2} \)
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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

−11.09689693856744265659419013991, −10.35687540277726914430138097298, −9.701177723743834316572896701440, −9.121315490798337839943671845377, −7.42781944412493334749344007291, −6.29192019014363472055821884258, −6.14737601887066146167647254934, −4.40377265890950208804967565889, −3.92105143892182480751337269234, −2.61437277649696332656153929787, 0.25792467019147517246779325662, 1.59526912977658231927452753996, 3.02636959412751566351169231344, 4.86606265651920213097689382783, 5.80234241687813084372302639302, 6.29929369962046180958725136370, 7.32960995832741241333539117798, 8.446287745649902042539963254591, 9.016082220493966625616937409533, 10.22963160441952311505697400051

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