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Properties

Label 1-388-388.131-r0-0-0
Degree $1$
Conductor $388$
Sign $-0.746 + 0.665i$
Analytic cond. $1.80186$
Root an. cond. $1.80186$
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  + (0.382 + 0.923i)3-s + (−0.195 + 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (−0.195 + 0.980i)19-s + (−0.831 + 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (−0.923 − 0.382i)27-s + (−0.555 − 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.195 + 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (−0.195 + 0.980i)19-s + (−0.831 + 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (−0.923 − 0.382i)27-s + (−0.555 − 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4779962024 + 1.253871173i\)
\(L(\frac12)\) \(\approx\) \(0.4779962024 + 1.253871173i\)
\(L(1)\) \(\approx\) \(0.9155920387 + 0.6896338542i\)
\(L(1)\) \(\approx\) \(0.9155920387 + 0.6896338542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.195 + 0.980i)T \)
7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (0.980 + 0.195i)T \)
19 \( 1 + (-0.195 + 0.980i)T \)
23 \( 1 + (0.555 + 0.831i)T \)
29 \( 1 + (-0.555 - 0.831i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (-0.555 + 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.195 + 0.980i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
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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

−23.99311665674876596516952408593, −23.48560539212476870687104440578, −22.849757388598917689022169160269, −21.16570631971381021726260395714, −20.3096806539935332574750082167, −20.133317561092512432896264226762, −18.89283987278813157080905070547, −18.0200377598080020556317947033, −17.1605695949893361597770690548, −16.382289176700823437805397524679, −15.18314615323046308959660445004, −14.2125967897107383130259019848, −13.14506513462210620054999420791, −12.84353108025847114691725005334, −11.72512801213735610913672978302, −10.661069279664996934510568769070, −9.38021498463565759083288679009, −8.466269775529752724126263282025, −7.60575358105353807345639397390, −6.86216793508697930255906553803, −5.48442469740134127480720071701, −4.40416671985284525527731849507, −3.23640992404060991092214219821, −1.73173970643785589092326518813, −0.790901093559745529471104436869, 2.02908779421757074676342483285, 3.25476434195179109187339103395, 3.76246124659393391508364525886, 5.47070966901575741434974221822, 5.96852494302959502624386124050, 7.61335198078831413273076379136, 8.46439982193706926548392813092, 9.349278828021588620765153271276, 10.49982301774948571860407835895, 11.094144735101684985863145565492, 12.031696343944441992191760650035, 13.51975994936658469599767214505, 14.35472920567511747316120074884, 15.09352584819204389610694251523, 15.84262707819819299836201235727, 16.61836396871417621679958794614, 17.96911820520331282025036682884, 18.92545185006624550160670532423, 19.268773855260343978974400172381, 20.947230443821962373366516780946, 21.16858516467337174726178831554, 22.11989112138775346346251903750, 22.89309267621767837610468391197, 23.81903901266557640382011417614, 25.20425886896187135583965744299

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