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Properties

Label 1-403-403.21-r0-0-0
Degree $1$
Conductor $403$
Sign $0.606 - 0.795i$
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  + (−0.951 + 0.309i)2-s + (0.978 − 0.207i)3-s + (0.809 − 0.587i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (0.994 − 0.104i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.951 − 0.309i)15-s + (0.309 − 0.951i)16-s + (−0.104 + 0.994i)17-s + (−0.743 + 0.669i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.978 − 0.207i)3-s + (0.809 − 0.587i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (0.994 − 0.104i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.951 − 0.309i)15-s + (0.309 − 0.951i)16-s + (−0.104 + 0.994i)17-s + (−0.743 + 0.669i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.606 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034252683 - 0.5120228617i\)
\(L(\frac12)\) \(\approx\) \(1.034252683 - 0.5120228617i\)
\(L(1)\) \(\approx\) \(0.9416722202 - 0.1868731781i\)
\(L(1)\) \(\approx\) \(0.9416722202 - 0.1868731781i\)

Euler product

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

−24.73008009599447304466644527602, −24.03200154141600276370884701044, −22.33540848200173387714064608542, −21.81088289885068913402561688907, −20.71514008571428176790206928705, −19.86633917091333072306414209029, −19.45183966438534920827836905860, −18.38682575006202161711030539345, −17.97062309963200446770418473981, −16.38342835531982683203258470058, −15.72466981261766568006540461482, −14.95321449359652427753989885649, −14.12964798645028114518312720260, −12.66028610674815493433462401872, −11.6580328937018190455949019944, −11.151528431993214847306658558268, −9.67355971231796444348166842676, −9.18736454532026418382667612776, −8.16058558869616262015094751676, −7.50040195444187313941175119068, −6.50198712643102319700506213480, −4.64274069533593603596670704650, −3.39512537523102758733628244986, −2.694563026712871443773944805849, −1.46141980608193337686590431350, 0.93463443918950066090710135344, 1.89592023750369709312412757730, 3.56976998933155845204496786573, 4.3428582090852777029828058685, 6.11582020390422282861667771984, 7.28776700443945879996762285690, 7.88554961734097466866717072124, 8.61030096004525513603275742570, 9.53127252037012802617801394535, 10.52678562317761489148726417701, 11.618564202355692246291726552600, 12.52795967984772759647867264906, 13.91206971431695399583532414349, 14.61202281726234837591240033448, 15.46998792693172517476985454340, 16.43225974574651460391304307842, 17.138047130058687544412493698186, 18.2210687053940808851349867892, 19.2268585768121197163282126728, 19.77867568917644764227057987276, 20.37061220828934614131825775680, 21.12411308324669186026565949099, 22.73992622115400496126051359005, 23.92578849853187952457322658272, 24.279019888827577860412259094627

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