Properties

Label 1-592-592.51-r0-0-0
Degree $1$
Conductor $592$
Sign $-0.980 - 0.194i$
Analytic cond. $2.74923$
Root an. cond. $2.74923$
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  + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s i·11-s + (−0.5 + 0.866i)13-s + (−0.866 + 0.5i)15-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s i·23-s + (−0.5 − 0.866i)25-s i·27-s − 29-s i·31-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s i·11-s + (−0.5 + 0.866i)13-s + (−0.866 + 0.5i)15-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s i·23-s + (−0.5 − 0.866i)25-s i·27-s − 29-s i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.980 - 0.194i$
Analytic conductor: \(2.74923\)
Root analytic conductor: \(2.74923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 592,\ (0:\ ),\ -0.980 - 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04017505393 - 0.4098410829i\)
\(L(\frac12)\) \(\approx\) \(0.04017505393 - 0.4098410829i\)
\(L(1)\) \(\approx\) \(0.6281839341 - 0.2240562852i\)
\(L(1)\) \(\approx\) \(0.6281839341 - 0.2240562852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.15933245622598562547559569729, −22.55018287776002264299580776861, −22.29762238065587107408349130504, −21.08294489565458700846513977541, −20.34696372848686645302693160417, −19.4353517617735521054204613511, −18.18016140137226672510410506116, −17.67697949265261613940929281253, −17.03171229636085648325621941949, −15.975560565219916669641309794557, −15.24042334288361856207720611788, −14.37098132013020517323824584239, −13.3325328679703178106684854904, −12.48452402675133418082054890771, −11.43768247011158853955044265443, −10.54861474461944098716990779124, −9.99642522051389685362585476671, −9.35408441398957551543311659816, −7.44105554975323316182573201929, −7.02059940715729781339242319136, −5.954659959186352021122442444431, −5.08628608451059759256687991615, −3.95916422843286623200359708347, −3.03954781409166457433887989305, −1.56151420471413117183009658454, 0.22878554356141947538921956163, 1.66840645451741902949809046593, 2.619470488851712421906502706949, 4.33552446769515454565557864890, 5.220677061855954881913811141489, 6.07750231438391350214424370627, 6.67560686574617863994382444858, 8.06378211663008860961318008508, 9.0030792674717560873695866786, 9.70293931145670824659951707764, 11.05525445969978774362766195019, 11.6787797804892840731786491173, 12.68907758573689762778787726579, 13.13982884208017164288568637037, 14.08351704199557390236172709906, 15.44622651016506782953300890426, 16.32300955444684058803636223180, 16.83104459213826899326589318728, 17.69260490695214439817861320989, 18.6226046509082766352104077327, 19.21537747226737265036081471354, 20.21334252445131119925580270113, 21.45672952343011882594758089601, 21.91596061660897552676828297044, 22.5594256472436402621589515914

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