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Properties

Label 1-403-403.228-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.976 - 0.213i$
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  + (−0.406 + 0.913i)2-s + (−0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (−0.866 − 0.5i)5-s − i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.669 − 0.743i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (0.994 + 0.104i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (0.406 + 0.913i)18-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (−0.866 − 0.5i)5-s − i·6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (0.669 − 0.743i)9-s + (0.809 − 0.587i)10-s + (0.951 + 0.309i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (0.994 + 0.104i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (0.406 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.976 - 0.213i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.976 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01494267684 + 0.1383047327i\)
\(L(\frac12)\) \(\approx\) \(0.01494267684 + 0.1383047327i\)
\(L(1)\) \(\approx\) \(0.3871011520 + 0.1705534896i\)
\(L(1)\) \(\approx\) \(0.3871011520 + 0.1705534896i\)

Euler product

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

−23.46310515858731445173883196418, −22.818567025844086785874041617383, −22.29766671779684646170827149635, −21.48548477785527931828360820695, −20.15861369147347281713889413605, −19.26780637025604063598856515715, −18.83490752165019904131169341906, −18.10609851757972382638454699242, −16.796796980111211137955847235995, −16.45682385967992854272195380524, −15.179565579082244079272699319042, −13.8064752592700823717046178017, −12.84739000793365502023803808476, −11.93602306001204770252841922533, −11.567062161680571530943247439665, −10.54336043647127670091202168786, −9.65194349175748274330687922918, −8.51911691490987344016443048282, −7.32670473440531107674769647064, −6.59214491016482444253024954022, −5.224976324338736261205687426892, −3.88574180764561693112198964421, −3.10297807084533809417317085948, −1.587980932461119975134959115150, −0.12918789005773170904626943703, 1.15594191022578475396845134588, 3.77302511343136515260258463932, 4.39339041976023677318592470557, 5.532598678488454699103007312088, 6.60510655180852812283563501245, 7.14975837426154538888889922137, 8.56491375812295421002174126035, 9.35730257763912599814035826075, 10.312134011117664920278960688365, 11.24905178109193772007854665020, 12.45602429077600254446203780474, 13.11773509433335215901589032235, 14.6886977920276827659622735336, 15.38776522802582580771070768296, 16.164819641542745191950522949131, 16.99152286516472211183594819212, 17.25731676737811342571038546276, 18.7341191319896673715980094446, 19.3852874970405353071770271620, 20.25578134695629792529427465558, 21.66248058439361373126075873521, 22.72808264371388714946778818210, 23.00226109464530213466424477724, 23.961251821854888463543356496687, 24.61070477885173795830271231726

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