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Properties

Label 1-368-368.213-r0-0-0
Degree $1$
Conductor $368$
Sign $0.794 - 0.607i$
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  + (0.540 − 0.841i)3-s + (0.909 + 0.415i)5-s + (0.959 + 0.281i)7-s + (−0.415 − 0.909i)9-s + (−0.755 − 0.654i)11-s + (0.281 + 0.959i)13-s + (0.841 − 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.989 + 0.142i)19-s + (0.755 − 0.654i)21-s + (0.654 + 0.755i)25-s + (−0.989 − 0.142i)27-s + (−0.989 + 0.142i)29-s + (0.841 − 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (0.909 + 0.415i)5-s + (0.959 + 0.281i)7-s + (−0.415 − 0.909i)9-s + (−0.755 − 0.654i)11-s + (0.281 + 0.959i)13-s + (0.841 − 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.989 + 0.142i)19-s + (0.755 − 0.654i)21-s + (0.654 + 0.755i)25-s + (−0.989 − 0.142i)27-s + (−0.989 + 0.142i)29-s + (0.841 − 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.794 - 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.823198672 - 0.6176124333i\)
\(L(\frac12)\) \(\approx\) \(1.823198672 - 0.6176124333i\)
\(L(1)\) \(\approx\) \(1.476732414 - 0.3210990831i\)
\(L(1)\) \(\approx\) \(1.476732414 - 0.3210990831i\)

Euler product

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

−24.84231992938605068648233083778, −24.06991590386397149578449376707, −22.84206219732324086016137292147, −21.93325917033093366475488981594, −21.02879662552742740058414675107, −20.59149771809245226471064158875, −19.88388338347726690424058515174, −18.40312138279119598101723114387, −17.54403521906673496108765011575, −16.90125351949482376101246062473, −15.62796871975448646836912314660, −15.05203871271963815638391331877, −13.941555603386238999053568899445, −13.36533959770827093032626564847, −12.17479198749245921198350464769, −10.58890262771962329664005847325, −10.39097879108024202036175715488, −9.16369345254860479645233187210, −8.32154858823081891575570152677, −7.42079162557472312079774588527, −5.611582364991153952629117506914, −5.08763934984489044191301506834, −3.96687272125667259283363042716, −2.62329650957538443965593071554, −1.554625811983267957523108040169, 1.35452331141883108450744434345, 2.29224047158206023492407634365, 3.242012668879472357454408928672, 4.98605673638018873949492081442, 5.957130189907070116209885407667, 6.98938333435568718586323955012, 7.92867741032317433814361811112, 8.89345159632798796666131082622, 9.774011504835622027064414399825, 11.179791838883050915606079281473, 11.800544317526891088056607218479, 13.194447233536391614519388689430, 13.84348152012986341354900049801, 14.38328790223233385542800712530, 15.486308164166907548037651740, 16.76118274153314884863789613053, 17.82468652090658798025744773368, 18.45757286248025294892960982991, 18.90273659955042561606524561391, 20.410343117899036736977650698484, 20.93634214484964857638558060370, 21.78853088895333118191716618022, 22.91059950416339267796152970680, 24.02459871446900049807364466631, 24.52790801782543672528608738677

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