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Properties

Label 1-407-407.54-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.319 - 0.947i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s − i·6-s + (−0.173 − 0.984i)7-s + (0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s + (−0.866 + 0.5i)14-s + (−0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s − i·6-s + (−0.173 − 0.984i)7-s + (0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s + (−0.866 + 0.5i)14-s + (−0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.319 - 0.947i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.319 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6021287078 - 0.8385753563i\)
\(L(\frac12)\) \(\approx\) \(0.6021287078 - 0.8385753563i\)
\(L(1)\) \(\approx\) \(0.8250991507 - 0.4405914386i\)
\(L(1)\) \(\approx\) \(0.8250991507 - 0.4405914386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (0.984 - 0.173i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−24.61857796897469071932302943195, −23.94135937235168375610890210043, −23.29470775750083589375126366752, −22.0406446126140617702489115675, −21.17295037453301982433096326772, −19.703695816610552549088908647452, −19.30680996195380451721378012620, −18.69029497130345665333755616175, −17.6543570980718074702562075176, −16.48760633995389304689845955261, −15.66970425117348562436412696945, −14.903352490315790712807174454609, −14.45324516391884373039920415745, −13.05782808039136111260535885092, −12.447551834397255698764554856307, −11.16174123174550646192183151176, −9.620242400048974801039509067509, −8.96906409501363716814092994037, −8.23093322438536734800819750228, −7.34544787904987460072048918672, −6.56516909410408109585051520638, −5.17128043686000762312092384557, −4.12763858372394336221357066671, −2.910199628588235161564663221760, −1.39762815460089539918685026849, 0.6678196410910188899848100343, 2.31673584869911212415305874675, 3.39939878017815690677169852047, 3.94744625437778684412970907655, 4.9913315233720705152729275438, 7.30213186266211993549410902389, 7.72670717382649967829593167631, 8.688500483248781235031808025870, 9.85064426290961818548333485154, 10.4072975442866366607394865012, 11.38395735800775668476584353149, 12.50758927301717601353918841391, 13.27360255471764521361764546543, 14.31783507527308043483927032695, 15.05249108145019796762250921592, 16.33418073456065920040321060609, 16.97308846701512501620671788271, 18.39278803264447259859058277688, 19.15130467002468874665070537896, 19.72357688854888216114550813583, 20.63307907554345608152603705716, 20.89434382412610249444612659505, 22.51525381539176811102111317604, 22.73527819194490510204561453244, 23.99312052759588851459278387371

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