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Properties

Label 1-353-353.76-r0-0-0
Degree $1$
Conductor $353$
Sign $0.259 + 0.965i$
Analytic cond. $1.63932$
Root an. cond. $1.63932$
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)2-s + (0.681 + 0.731i)3-s + (0.654 − 0.755i)4-s + (−0.731 + 0.681i)5-s + (−0.923 − 0.382i)6-s + (0.923 − 0.382i)7-s + (−0.281 + 0.959i)8-s + (−0.0713 + 0.997i)9-s + (0.382 − 0.923i)10-s + (0.540 − 0.841i)11-s + (0.999 − 0.0356i)12-s + (0.968 + 0.247i)13-s + (−0.681 + 0.731i)14-s + (−0.997 − 0.0713i)15-s + (−0.142 − 0.989i)16-s + (0.281 − 0.959i)17-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (0.681 + 0.731i)3-s + (0.654 − 0.755i)4-s + (−0.731 + 0.681i)5-s + (−0.923 − 0.382i)6-s + (0.923 − 0.382i)7-s + (−0.281 + 0.959i)8-s + (−0.0713 + 0.997i)9-s + (0.382 − 0.923i)10-s + (0.540 − 0.841i)11-s + (0.999 − 0.0356i)12-s + (0.968 + 0.247i)13-s + (−0.681 + 0.731i)14-s + (−0.997 − 0.0713i)15-s + (−0.142 − 0.989i)16-s + (0.281 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(353\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(1.63932\)
Root analytic conductor: \(1.63932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{353} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 353,\ (0:\ ),\ 0.259 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9015399777 + 0.6914463300i\)
\(L(\frac12)\) \(\approx\) \(0.9015399777 + 0.6914463300i\)
\(L(1)\) \(\approx\) \(0.8601586722 + 0.4077874817i\)
\(L(1)\) \(\approx\) \(0.8601586722 + 0.4077874817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad353 \( 1 \)
good2 \( 1 + (-0.909 + 0.415i)T \)
3 \( 1 + (0.681 + 0.731i)T \)
5 \( 1 + (-0.731 + 0.681i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (0.968 + 0.247i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.599 + 0.800i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
29 \( 1 + (-0.755 + 0.654i)T \)
31 \( 1 + (0.570 + 0.821i)T \)
37 \( 1 + (0.994 - 0.106i)T \)
41 \( 1 + (-0.977 + 0.212i)T \)
43 \( 1 + (0.977 - 0.212i)T \)
47 \( 1 + (-0.997 + 0.0713i)T \)
53 \( 1 + (0.0356 - 0.999i)T \)
59 \( 1 + (-0.923 + 0.382i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (0.948 + 0.315i)T \)
73 \( 1 + (-0.755 - 0.654i)T \)
79 \( 1 + (0.731 + 0.681i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (-0.106 + 0.994i)T \)
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

−24.687337762079624771877854235266, −24.09747020952543738161597803041, −23.08453045321113401406844923724, −21.55088665203729775007830316152, −20.6546995384565248951485105395, −20.14873825134147458535970396876, −19.342570091358579445145924511102, −18.51573034420311121732175278169, −17.653086082258440300564739660821, −16.96721041336741804436205922693, −15.45521665721412936418989884350, −15.107292748477919350499372966677, −13.52735298243603177796109805641, −12.632426997611993180048360242914, −11.797244531263737193256841405427, −11.17009153633409142902646299723, −9.53220757054408758969987025245, −8.84643523559528542760887403283, −7.93766701275249696728819344581, −7.50061808064335884951360704426, −6.089195904857012367630472512541, −4.33250371861135250577418997650, −3.29346089069459072241566860, −1.855380821232115101468632206883, −1.13128985325892991413448052804, 1.29131281516138600477858720511, 2.85279908307397844889695963571, 3.88733201644044716536952164004, 5.16208360966211374325836575341, 6.54660155341185079725711684523, 7.66438366263756059709282665816, 8.303641827358681889137989810763, 9.13978553613313992594841105259, 10.3535534103213078731338457803, 11.048988226088896418239577340122, 11.69690835625500210225039674800, 13.923282494763432420092532058202, 14.35130452724188170754451602381, 15.1745882856634436482331064974, 16.25784231842954675616589980794, 16.57124975029562938165674841192, 18.143755269185174430188839789, 18.68747346624578549809296478701, 19.63272440187079880702247210473, 20.47466791987516624128638805173, 21.09946621035422146448508354805, 22.423667349074078477064811116733, 23.35466654404840490256338105700, 24.37333724421924269182840766380, 25.113230128274616943347423096281

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