Properties

Label 1-403-403.9-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.989 - 0.147i$
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  + (−0.104 + 0.994i)2-s + (−0.104 − 0.994i)3-s + (−0.978 − 0.207i)4-s + (−0.5 + 0.866i)5-s + 6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)11-s + (−0.104 + 0.994i)12-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)15-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + (−0.104 − 0.994i)18-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.104 − 0.994i)3-s + (−0.978 − 0.207i)4-s + (−0.5 + 0.866i)5-s + 6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)11-s + (−0.104 + 0.994i)12-s + (−0.978 + 0.207i)14-s + (0.913 + 0.406i)15-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + (−0.104 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.989 - 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03180079663 + 0.4276609149i\)
\(L(\frac12)\) \(\approx\) \(-0.03180079663 + 0.4276609149i\)
\(L(1)\) \(\approx\) \(0.5659880706 + 0.3379155709i\)
\(L(1)\) \(\approx\) \(0.5659880706 + 0.3379155709i\)

Euler product

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

−23.48225415892892739212435866783, −23.071607619855972635624200813006, −21.79575376510792483761826763204, −21.24410586868584632926196682553, −20.46677103319735188995468479289, −19.7862834354692498833491970547, −19.07546330563271658721586426863, −17.597768340063454331824802184349, −16.82157592298360651057975443932, −16.36247378343706935738771669254, −14.95029347911671592659378583757, −14.06196556128636618951525140708, −13.1268984140373005670023658518, −12.01835131917815726711561253569, −11.26444305572193188454872144819, −10.44171790131958359953167147017, −9.67045753056790694062576009769, −8.433173303325498111777894536075, −8.12777991550810561269714386097, −6.00145552872613933063933161354, −4.85493155464601861117877511546, −3.97645509661355245585638839559, −3.52649132727243363971114128661, −1.68252365611185411394468080577, −0.26643109403937925140027665973, 1.773765075259592803944277525583, 3.052749708458097931994200575339, 4.60558964263528282476295706352, 5.68878504429936899618868090960, 6.69881428923520385291974511029, 7.264711961881482527556961742356, 8.19971040727226871809568232356, 9.074923227153584028104173308639, 10.351019878141676170081707768875, 11.71376413448873446587726802300, 12.29801879369959132978003206211, 13.483049324612378535969466593705, 14.4278913722776352867675402683, 15.0237134514750412985979924126, 15.87979208321032581278735597973, 17.118538545890183568063813556522, 17.99633427778995450220252695841, 18.432827089229833214152601838342, 19.20272714225074548045038645009, 20.15828073830506106820445855832, 21.84894265919945151618203137632, 22.47813019995034964092890675555, 23.28032611393759798891035354752, 23.9568627231221559992656347464, 24.84058404771390913088732383900

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