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Properties

Label 1-403-403.35-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.507 + 0.861i$
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  + (0.913 + 0.406i)2-s + (−0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.913 + 0.406i)10-s + (−0.978 + 0.207i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (−0.104 + 0.994i)15-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.913 + 0.406i)10-s + (−0.978 + 0.207i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (−0.104 + 0.994i)15-s + (−0.104 + 0.994i)16-s + (−0.978 − 0.207i)17-s + (−0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257921757 + 2.199594329i\)
\(L(\frac12)\) \(\approx\) \(1.257921757 + 2.199594329i\)
\(L(1)\) \(\approx\) \(1.500974733 + 1.225527479i\)
\(L(1)\) \(\approx\) \(1.500974733 + 1.225527479i\)

Euler product

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

−24.02864089928225827069383779303, −23.237833692999494427429008988269, −22.49699038391881196618228360771, −21.27976929225861396379975479522, −20.83818964232713578962514468658, −19.84795125325430495460081646705, −18.92510996572408603589164265977, −17.95476432549610217618828064203, −17.28462940095312358078835560830, −16.13526605919145483113184342448, −14.7813300820247192277745738918, −14.025254838125025114925583431797, −13.31481066293003360073345877450, −12.84460793358657871386120409595, −11.58786352962636726783719785761, −10.80962519522390020676520831227, −9.98153920691107677617711650005, −8.42178608564810470299085840892, −7.32542093297405565440509521198, −6.38782067038030385934206441796, −5.53285773606945826935244028088, −4.62375269967154424035534050619, −3.055815329940128421918750393279, −2.04848243186572030724554685722, −1.196075314312932836847299086550, 2.27221787913938017026714546049, 2.86491673955307379070833022571, 4.54716766441387014147513085259, 5.05722599681253326322285011440, 5.85959223063384281892484615847, 6.94356828048943336953045138456, 8.449283502988252806153627749570, 9.11609282633843033792049169054, 10.534543617139827860607890365643, 11.109111921329869007223864255169, 12.29304303597558965186065973567, 13.2843139121799806162326902648, 14.15299979535276606509917132092, 15.03567349223321087085622076219, 15.60969320184011315930775660610, 16.537507758935734937293675382430, 17.58374180224148960707870401653, 18.063065467608462400697322326485, 19.86501383393144292072586316112, 20.91465887832283053475198400808, 21.23157810739682648340878629774, 21.995868262662606550716339786471, 22.67306643830144298301746648223, 23.76438989687405057611692487133, 24.65799219250687917214609495343

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