Properties

Label 1-368-368.317-r0-0-0
Degree $1$
Conductor $368$
Sign $0.955 - 0.293i$
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  + (0.989 − 0.142i)3-s + (−0.281 + 0.959i)5-s + (0.654 − 0.755i)7-s + (0.959 − 0.281i)9-s + (−0.540 − 0.841i)11-s + (0.755 − 0.654i)13-s + (−0.142 + 0.989i)15-s + (0.415 − 0.909i)17-s + (−0.909 + 0.415i)19-s + (0.540 − 0.841i)21-s + (−0.841 − 0.540i)25-s + (0.909 − 0.415i)27-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.654 − 0.755i)33-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (−0.281 + 0.959i)5-s + (0.654 − 0.755i)7-s + (0.959 − 0.281i)9-s + (−0.540 − 0.841i)11-s + (0.755 − 0.654i)13-s + (−0.142 + 0.989i)15-s + (0.415 − 0.909i)17-s + (−0.909 + 0.415i)19-s + (0.540 − 0.841i)21-s + (−0.841 − 0.540i)25-s + (0.909 − 0.415i)27-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.654 − 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.955 - 0.293i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.955 - 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.858084689 - 0.2791388590i\)
\(L(\frac12)\) \(\approx\) \(1.858084689 - 0.2791388590i\)
\(L(1)\) \(\approx\) \(1.485952097 - 0.09712531725i\)
\(L(1)\) \(\approx\) \(1.485952097 - 0.09712531725i\)

Euler product

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

−24.7607326368681378359036528198, −23.94317189796259519234854284559, −23.24482641135488993149146719456, −21.555742797909013088065967203, −21.19108299764750644687487446657, −20.42195113650981024711919810607, −19.498366784817726567176625122012, −18.71143913443524048225653350721, −17.69538317608048689025343639603, −16.587198062622436088818878279871, −15.517578649579397626399714863494, −15.08785087362456098807775230501, −13.97923404171277652354491831396, −12.93580514089540348177192261242, −12.32674780826890679322575494465, −11.096838385949853512734462400869, −9.87130656677956037231945342495, −8.85329031938267369523678154667, −8.354819952469193218749699577675, −7.44235242902007444087940526502, −5.89749196886721712142709954989, −4.6368163720529634465749651381, −4.01499257778234678924018452147, −2.40402377343235853935915860409, −1.566751358402897876443937099173, 1.21791118168433388956079703205, 2.75389598108651698306885736706, 3.42391180055845343441022575941, 4.56458612398502919851739732738, 6.12779330548364783058789086774, 7.2616934484372495281805979869, 7.96495846244908119670288696945, 8.755092711961408067816580496723, 10.32393739625841237971522527893, 10.67941027224484027121984970706, 11.92272561529364051585544409713, 13.28954647897153346267958053662, 13.92397784153034345858433026186, 14.65060526002328521358435522135, 15.53531999711945112199040380201, 16.45172916421527784275329295543, 17.93666243446823721717237251718, 18.45595350975451988754608803053, 19.34033655680719034136469933447, 20.23076029135631304906421804039, 21.03474705472565794779624142440, 21.77410929440709984949428715182, 23.23268661004671980872687755941, 23.53005155835555442785175327849, 24.77670400018696992762850749114

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