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Properties

Label 1-403-403.16-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.378 + 0.925i$
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  + (0.669 + 0.743i)2-s + (−0.978 − 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (−0.978 − 0.207i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.978 − 0.207i)3-s + (−0.104 + 0.994i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (0.669 + 0.743i)10-s + (0.913 − 0.406i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (−0.978 − 0.207i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8895137622 + 1.324146221i\)
\(L(\frac12)\) \(\approx\) \(0.8895137622 + 1.324146221i\)
\(L(1)\) \(\approx\) \(1.083725149 + 0.7282901821i\)
\(L(1)\) \(\approx\) \(1.083725149 + 0.7282901821i\)

Euler product

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

−23.72220339536268906702610599001, −23.01201130442486105623872718463, −22.41669894509456917308435398703, −21.41160979367367485538989924573, −21.00529234925748556692067046522, −19.897827293611612230257120406504, −18.97944496757541962793407071712, −17.789184823338296119618338097204, −17.24006107995372203475216272210, −16.31849815472169897411602759354, −15.05697669814040423439201667040, −14.1036416970544107690918732254, −13.35456130380262266599297951938, −12.42936854791709686465261957770, −11.59330309223365023390859930003, −10.564439570826404687806691037566, −10.01437742437406755727945304453, −9.22153256789298056133818419979, −7.116838139555409406959985186489, −6.32498831804302660288129536699, −5.43079118613831073701632076312, −4.4599274531133213985259780592, −3.56960881102639389697541899823, −1.92632638449543084755549515076, −0.92741600721119290747256671017, 1.620762809164109197589499767678, 2.9825097765484389667545220124, 4.48524579252931165163954520206, 5.41063400259414445347920686794, 6.27392525518659974915650472428, 6.557159635683218416448199456918, 8.152720215918509361691068865152, 9.100861404930292037845924220299, 10.2858947427045178014857498245, 11.49905860426806142653158084057, 12.43569105751382277560500692669, 12.88396638369699746280207785791, 14.13897292019953759482994644037, 14.79883309986148208575331730985, 16.00500512943025093960454539716, 16.77775707725714157599885629556, 17.352335184714488605549630450, 18.284832312952321313351773592250, 19.0390465720306557885626473342, 20.78686925648126579529034241185, 21.75359682765840164491216594281, 21.91428881727110682260414983012, 22.86992034191423051914702760582, 23.773346181651033143539813116994, 24.70818507091606771548901198547

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