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Properties

Label 1-229-229.31-r1-0-0
Degree $1$
Conductor $229$
Sign $0.252 + 0.967i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.475 − 0.879i)2-s + (−0.962 + 0.272i)3-s + (−0.546 + 0.837i)4-s + (−0.975 + 0.218i)5-s + (0.697 + 0.716i)6-s + (−0.771 − 0.635i)7-s + (0.996 + 0.0825i)8-s + (0.851 − 0.523i)9-s + (0.656 + 0.754i)10-s + (−0.789 − 0.614i)11-s + (0.298 − 0.954i)12-s + (0.735 − 0.677i)13-s + (−0.191 + 0.981i)14-s + (0.879 − 0.475i)15-s + (−0.401 − 0.915i)16-s + (−0.677 − 0.735i)17-s + ⋯
L(s)  = 1  + (−0.475 − 0.879i)2-s + (−0.962 + 0.272i)3-s + (−0.546 + 0.837i)4-s + (−0.975 + 0.218i)5-s + (0.697 + 0.716i)6-s + (−0.771 − 0.635i)7-s + (0.996 + 0.0825i)8-s + (0.851 − 0.523i)9-s + (0.656 + 0.754i)10-s + (−0.789 − 0.614i)11-s + (0.298 − 0.954i)12-s + (0.735 − 0.677i)13-s + (−0.191 + 0.981i)14-s + (0.879 − 0.475i)15-s + (−0.401 − 0.915i)16-s + (−0.677 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07333875746 - 0.05667966314i\)
\(L(\frac12)\) \(\approx\) \(-0.07333875746 - 0.05667966314i\)
\(L(1)\) \(\approx\) \(0.3217080952 - 0.2091697109i\)
\(L(1)\) \(\approx\) \(0.3217080952 - 0.2091697109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.475 - 0.879i)T \)
3 \( 1 + (-0.962 + 0.272i)T \)
5 \( 1 + (-0.975 + 0.218i)T \)
7 \( 1 + (-0.771 - 0.635i)T \)
11 \( 1 + (-0.789 - 0.614i)T \)
13 \( 1 + (0.735 - 0.677i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
19 \( 1 + (-0.298 - 0.954i)T \)
23 \( 1 + (0.656 - 0.754i)T \)
29 \( 1 + (-0.936 + 0.350i)T \)
31 \( 1 + (-0.376 - 0.926i)T \)
37 \( 1 + (0.993 - 0.110i)T \)
41 \( 1 + (-0.999 - 0.0275i)T \)
43 \( 1 + (-0.401 + 0.915i)T \)
47 \( 1 + (-0.523 - 0.851i)T \)
53 \( 1 + (0.245 - 0.969i)T \)
59 \( 1 + (-0.110 + 0.993i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (0.999 - 0.0275i)T \)
71 \( 1 + (-0.137 - 0.990i)T \)
73 \( 1 + (0.426 + 0.904i)T \)
79 \( 1 + (-0.936 - 0.350i)T \)
83 \( 1 + (-0.592 + 0.805i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.821 - 0.569i)T \)
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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

−26.9264939183126735751932688270, −25.88872057944739396233948513207, −25.00463604090810086774024451191, −23.89837398422850028172168429864, −23.34968079557855734920494032230, −22.72499340403741042265563496857, −21.57157621279918191943765561435, −20.00540434846757127104200224558, −18.79944486140681981143904123651, −18.61426739933145292807666145735, −17.29734849017826114837474235374, −16.40346788398439803766012261598, −15.71507780506789224783002585546, −15.047898681463384025454898552296, −13.28573202642694359481909624144, −12.56564588089856295124475448700, −11.350168769728458687649395619908, −10.36970601087102943292934020383, −9.14119997015391145659796014617, −8.0398868510901058796863739438, −7.035590284166532657526924708504, −6.137990162869525522345858230028, −5.110136838565786494080084061879, −3.94467585302450675897974237368, −1.61385998101681029496348462001, 0.073232220232927902245509386539, 0.6587814103456342027440990763, 2.9245418379710795218682375654, 3.875179053045470074188089449846, 4.955869644021685304497875827325, 6.6193101061205615536011827413, 7.62974273112414295651583285343, 8.88984329456488049334813838932, 10.13586095462138003482596767799, 11.03484885035506121561824507661, 11.37653559745235538252543367077, 12.85870779496766435227874625220, 13.26355926878307981821466317, 15.28216412040289281147279820548, 16.21199854662157362657011908088, 16.82846583164546562327482228613, 18.185160451223915390709281309335, 18.66423010615850321363442548970, 19.86679910811827708670102948917, 20.57089271831849637714002235025, 21.74528939455380311144758658529, 22.69300353121651836499534517142, 23.11961142929325241342453920484, 24.174552630441346972962395605547, 25.94295360038206793421025977597

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