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Properties

Label 1-368-368.13-r0-0-0
Degree $1$
Conductor $368$
Sign $0.828 - 0.560i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (−0.909 + 0.415i)3-s + (−0.755 + 0.654i)5-s + (−0.841 − 0.540i)7-s + (0.654 − 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.540 + 0.841i)13-s + (0.415 − 0.909i)15-s + (−0.959 + 0.281i)17-s + (0.281 − 0.959i)19-s + (0.989 + 0.142i)21-s + (0.142 − 0.989i)25-s + (−0.281 + 0.959i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (−0.755 + 0.654i)5-s + (−0.841 − 0.540i)7-s + (0.654 − 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.540 + 0.841i)13-s + (0.415 − 0.909i)15-s + (−0.959 + 0.281i)17-s + (0.281 − 0.959i)19-s + (0.989 + 0.142i)21-s + (0.142 − 0.989i)25-s + (−0.281 + 0.959i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4764700828 - 0.1461171517i\)
\(L(\frac12)\) \(\approx\) \(0.4764700828 - 0.1461171517i\)
\(L(1)\) \(\approx\) \(0.5840206173 + 0.04662427830i\)
\(L(1)\) \(\approx\) \(0.5840206173 + 0.04662427830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.909 + 0.415i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (-0.989 + 0.142i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (-0.281 - 0.959i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (-0.540 - 0.841i)T \)
61 \( 1 + (0.909 + 0.415i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−24.65185923335019877959431460546, −23.75140767275826018211851354896, −23.00375283696106239480152724473, −22.43098624432493565332200276081, −21.30621051909891757737516490865, −20.27810141818814198153179245969, −19.3667743608871899583094689186, −18.47440046522313578529493327898, −17.83136278073644155272877107515, −16.538250161177636793299842993742, −15.95263175547006870896550595029, −15.3996869157488843200144426421, −13.638587208718554129730177609777, −12.66417413059799956710957351837, −12.41013688401686990818338024880, −11.164283460589005722918590146920, −10.41962821547911536122055275685, −9.07587618074278384176809797965, −8.04668600587618530270457593795, −7.13869435833218593959751459383, −5.88055512808146459977766032011, −5.25501772710004614567388208207, −3.97639283294875642912161014684, −2.62602547111475443558129795905, −0.93576628678089292550153593666, 0.45214139990415070001932503744, 2.59923299927608877924151377141, 3.893398803076756443453174885931, 4.56210753240545286352271914970, 6.07368202012419492852591231324, 6.776678184741127947342464069307, 7.70957579358193812788483378785, 9.21143428859330933034602722249, 10.19173259962323102256293806158, 11.03070892369973488585145735497, 11.60987514772444847060865904424, 12.85898746008533360734619129492, 13.64244210390245267888568706840, 15.17473420334572365478508675900, 15.706025062075712456358618622191, 16.43704726929332118803691002684, 17.4701151675051528579266520903, 18.41118831852214466736872919816, 19.17258494373809785169995262175, 20.19817442559599113086214125195, 21.20724534818731012276597625790, 22.236198008132562134847108316272, 22.75532221042004620540838477654, 23.65532886441252333760658900782, 24.06568498998402598650163097661

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