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Properties

Label 1-368-368.29-r0-0-0
Degree $1$
Conductor $368$
Sign $0.991 + 0.131i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.991 + 0.131i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.991 + 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9777818114 + 0.06470560259i\)
\(L(\frac12)\) \(\approx\) \(0.9777818114 + 0.06470560259i\)
\(L(1)\) \(\approx\) \(0.8704542272 + 0.1044874792i\)
\(L(1)\) \(\approx\) \(0.8704542272 + 0.1044874792i\)

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

−24.2945088571122164630719321965, −23.801290896176468787993799171094, −23.19024966311893413633160817691, −22.06291928263629727069140492246, −21.35174214542729026637248940186, −19.91196664484248882871768206009, −19.23303634632146813034915980757, −18.65291717856883038358595432812, −17.434325264585090260305155854692, −16.941737518767430188322500463289, −15.82496528258180113322149411572, −14.524377541942705823563807101251, −14.096407752100175726757724867121, −12.747653500587497158217292691804, −11.81159571603268346088256247516, −11.26994851236394412444589224650, −10.4545597472666042855425684815, −8.59407036921526119589518964609, −8.05276745201039165588449575892, −6.89296613172190421630456012425, −6.28038082721795341385730738638, −4.7741017051922661943877488224, −3.8901119854627403353835290320, −2.27492048132008064801747935709, −1.068119387871828083568730533904, 0.848951048234212514263272904703, 2.716155700203391684587601638, 4.23890593879418956002566241715, 4.63731091881222960596550025806, 5.73529571412202441709878973770, 7.10140812147923619353614669514, 8.221720313721240076363641192962, 9.068996279019736965120437389725, 10.17621269492394315221106119677, 11.2172350062028058434451048749, 11.8621592062142989637059474903, 12.61486552223412426903344201781, 14.22708805130819031089713521165, 15.21260576739843907150659164084, 15.526939237529879377071506359928, 16.78480913888671627420230214183, 17.41084051682281362528174687634, 18.315124065213844003526904264743, 19.68591980200553822329994851762, 20.37286917762511096884642560325, 21.11524128863804542075467157200, 22.14207280601755992056519916422, 22.929662704101520180914021788310, 23.60886814465717575216677691483, 24.65762992493224946499148033207

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