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Properties

Label 1-403-403.123-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.999 + 0.0386i$
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 − i·5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − i·5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.999 + 0.0386i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.999 + 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006396726089 + 0.3312773794i\)
\(L(\frac12)\) \(\approx\) \(0.006396726089 + 0.3312773794i\)
\(L(1)\) \(\approx\) \(0.5282272219 + 0.2437084528i\)
\(L(1)\) \(\approx\) \(0.5282272219 + 0.2437084528i\)

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 \( 1 - iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 - T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
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

−24.14968168222193426933368833536, −22.9499622260942487761730991924, −22.11121318000676124030914154767, −21.203767950793438873757529917452, −20.085178239956886403172163405208, −19.42112647133857046590920420172, −18.604433775501939397172467135598, −18.25207858589635799027596230171, −17.34376278223198803725515889236, −15.84292590487157223277418239848, −15.436218553014915303829200278917, −13.795563519834015490120723867406, −13.332956839108983747235813198701, −12.074507297076215465654124496172, −11.477830599758383980856566392538, −10.277191531264666472892518346313, −9.4345234613084062096309778738, −8.49641833714867751353547021686, −7.436694508390586750219095278291, −6.856017393584114286168576777441, −5.775312343288820800680747822305, −3.45948079961631174329559270907, −2.86935551526354588410090547608, −2.041704054351559518743435706543, −0.22593877085561199576570636378, 1.62550637862030854313083736999, 3.04847405585755547070089212750, 4.44144333720345657871330651412, 5.30750624495496986565102163341, 6.4917439381715180577535102656, 7.83453620382241709830230918892, 8.44965230417017607966330830043, 9.48648030520617593296006656668, 10.05736467569617734237446575205, 10.85117263342882142173047743798, 12.35233112050402375915537281637, 13.406487055498602311229805858017, 14.3904111295659929414496825008, 15.540964857957435168457397006243, 16.03765132939418994578885304583, 16.69852195003106687474910168143, 17.55367733163648757658182750193, 18.75058068714792126179132260278, 19.80683620819615308954089470254, 20.236183729265445226381689228818, 20.93244683973813893235037864801, 22.18726637069124594954371401335, 23.24737105945635028197480097053, 24.083222790421595635761409020613, 25.05919195733203548012766526606

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