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Properties

Label 1-403-403.386-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.999 + 0.0252i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.104 + 0.994i)2-s + (−0.809 + 0.587i)3-s + (−0.978 − 0.207i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)15-s + (0.913 + 0.406i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.809 + 0.587i)3-s + (−0.978 − 0.207i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)15-s + (0.913 + 0.406i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.999 + 0.0252i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.999 + 0.0252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003455121267 + 0.2736271653i\)
\(L(\frac12)\) \(\approx\) \(0.003455121267 + 0.2736271653i\)
\(L(1)\) \(\approx\) \(0.4706219443 + 0.2527066559i\)
\(L(1)\) \(\approx\) \(0.4706219443 + 0.2527066559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.104 + 0.994i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−23.67929514416113373851897711989, −22.94720385584987713966778260598, −22.04876659301412467439381984089, −21.61468705381563071180034384433, −20.36632455751409582875335886828, −19.39127062431463004113593758671, −18.59279379397604056144114748911, −18.0063994312309437262044943212, −17.51463499090145187608291654590, −15.975800927573968503400490147078, −15.06891149294918002609280928007, −13.87096793996840764090496217476, −13.00318544098192651465395302734, −12.06310530669345468742456873134, −11.28461136391017147548206219473, −10.86976798290839180253325290295, −9.74663206301331974185974764669, −8.30802404852758548919102136188, −7.635733279223829874450331020442, −6.34329168917599122889846941962, −5.15763705609362594362524576951, −4.29358647409990548141600411021, −2.62049235628533354988928478953, −2.07560894622931102592007690907, −0.20586367362678211532281661382, 1.21820052342872519380310430671, 3.83441062235460727505679522029, 4.528960219909402299640571978977, 5.29760292902064640422371184326, 6.256906113820983383084671777485, 7.57144411151964915235334995659, 8.229285157194070118904375579490, 9.33512806069439724816250499428, 10.37079764781706992603962682105, 11.212584862408397669251799790888, 12.46781919351704390466202803341, 13.26950937906783135298628428659, 14.45178324589107453940482197146, 15.37937468256589939850931487133, 16.26480858664300862100478614971, 16.57842926217001377673314659278, 17.700593613669141996160848544803, 18.14584100740868142885557030191, 19.58269017547565647048893494356, 20.63141005818835472404263664600, 21.38981158900257105264578001393, 22.47972962988125960847486822002, 23.35026350046707266903618964028, 23.922108133175117088151840946651, 24.384536811623997304298521982822

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