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Properties

Label 1-403-403.305-r0-0-0
Degree $1$
Conductor $403$
Sign $0.960 - 0.279i$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + i·6-s + i·7-s + i·8-s + (−0.5 − 0.866i)9-s − 10-s − i·11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + 17-s + (0.866 + 0.5i)18-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + i·6-s + i·7-s + i·8-s + (−0.5 − 0.866i)9-s − 10-s − i·11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + 17-s + (0.866 + 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.960 - 0.279i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.960 - 0.279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210207733 - 0.1727433125i\)
\(L(\frac12)\) \(\approx\) \(1.210207733 - 0.1727433125i\)
\(L(1)\) \(\approx\) \(0.9994466705 - 0.03841036911i\)
\(L(1)\) \(\approx\) \(0.9994466705 - 0.03841036911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 - iT \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 - iT \)
43 \( 1 + T \)
47 \( 1 + iT \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + iT \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−25.02780878729123430848184887216, −23.42762859139204749955395622356, −22.39262697535406257200126154104, −21.29693910747496414148546112865, −20.87873985776805489523940920153, −20.144281705249447327648396313480, −19.484476547213704718241106712642, −18.190399172483834015191772729040, −17.267200836072893200749665670573, −16.708586154392044502867668215111, −15.929983727670667315013084019915, −14.661931024593124861988561027, −13.73937109141383498636152931515, −12.79910556545450200729111130150, −11.69296301339908485347534823962, −10.369993892028997433483393632878, −9.98176764178115880853352457739, −9.357655453777252307057791439273, −8.14267636404132276215781296458, −7.43449456128570995277923055607, −5.91066500134077271579371678338, −4.521114281925797635966989920268, −3.66394212081318092349786493802, −2.368432335934547310469920533577, −1.30411059330812577966300965791, 1.060296359505921708641990981586, 2.31097844897171320868033261686, 2.974085077921120761470139757099, 5.40447023134284940350417858648, 6.11855412529079304960552837586, 6.86948955344189719018380780197, 8.028272848472807625663039879785, 8.819689860452996619298105202285, 9.52266554606379552809890877894, 10.70190421157888874897543035593, 11.703964946789615083618182290807, 12.82613307194103957481561128334, 14.04079868924398025004263985611, 14.46798914436588550192583823938, 15.49774141879222511997948663786, 16.5796084715218130515866304248, 17.61033530066773015767146062361, 18.30588952571025039452068333594, 18.832877554188285335705061400781, 19.55408142969214632358045922034, 20.72888395156929707981450779821, 21.61101804010447084757580770361, 22.69777198872144466250676306494, 24.038009397559765719800340072915, 24.37601371418801240210813095861

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