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Properties

Label 1-403-403.165-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.0452 + 0.998i$
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 + (0.669 + 0.743i)3-s + (−0.104 − 0.994i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (0.669 + 0.743i)18-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.669 + 0.743i)3-s + (−0.104 − 0.994i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (0.669 − 0.743i)12-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + (0.669 + 0.743i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0452 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0452 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0452 + 0.998i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.0452 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8829214865 + 0.9237909013i\)
\(L(\frac12)\) \(\approx\) \(0.8829214865 + 0.9237909013i\)
\(L(1)\) \(\approx\) \(1.226394733 + 0.2128436120i\)
\(L(1)\) \(\approx\) \(1.226394733 + 0.2128436120i\)

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 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
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

−24.12580923099361300827692590581, −23.59316189016706568502601853583, −22.71822265515723432937334969100, −21.566637484578835227757719027062, −20.56101530626511353683412253270, −19.92743245170024666900128293395, −19.029374275990996847582091706165, −17.85649155183818853478911842709, −16.97910209611092505003049810436, −15.94326523238091098265220180422, −15.50655506253042006406289937571, −14.18591492683986268756499270294, −13.42823294260792793946430346927, −12.82575161819332239751888510183, −12.18666468836092515197867766412, −10.7955219147880058222718328833, −9.06664912105270366290965019515, −8.549768313792852331183649542935, −7.52974006484845239346144710997, −6.80692969486032815963597491027, −5.72922431238602414944206185446, −4.455089747906153967970617185879, −3.533519001818215962976844001771, −2.49865613078407673667241406174, −0.5158759304949038026160049038, 2.20996828252327676453083674009, 2.91966575231159957576824442773, 3.72837094390239097307838181352, 4.77698278763790283959453485163, 5.89057932193805892152370212338, 7.11618841883565191418614981297, 8.37271413033424319582741797486, 9.73153244904986840359291283525, 10.03882463427826735404189210024, 11.15689413732418743071564469375, 12.00119669513197760858545553197, 13.155839622429496073957886583149, 13.912626372243004438173370512689, 14.95347043859721580454557465874, 15.50320992940127049355593526546, 16.10452798432042311304496663376, 18.01578551860779761512943005417, 18.80799098355159766241028378994, 19.56121746503119583371867404551, 20.24312467134230777975003314689, 21.221472123443831120698389800702, 21.93194454746554376218737565023, 22.75204409474935340084344140480, 23.18937011657866236535128257297, 24.61235459539760762138002714707

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