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Properties

Label 1-373-373.123-r0-0-0
Degree $1$
Conductor $373$
Sign $0.916 - 0.400i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.664 − 0.747i)2-s + (0.997 + 0.0675i)3-s + (−0.117 − 0.993i)4-s + (−0.217 + 0.975i)5-s + (0.713 − 0.701i)6-s + (−0.758 + 0.651i)7-s + (−0.820 − 0.571i)8-s + (0.990 + 0.134i)9-s + (0.585 + 0.810i)10-s + (0.905 + 0.425i)11-s + (−0.0506 − 0.998i)12-s + (0.820 − 0.571i)13-s + (−0.0168 + 0.999i)14-s + (−0.283 + 0.959i)15-s + (−0.972 + 0.234i)16-s + (0.918 + 0.394i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.747i)2-s + (0.997 + 0.0675i)3-s + (−0.117 − 0.993i)4-s + (−0.217 + 0.975i)5-s + (0.713 − 0.701i)6-s + (−0.758 + 0.651i)7-s + (−0.820 − 0.571i)8-s + (0.990 + 0.134i)9-s + (0.585 + 0.810i)10-s + (0.905 + 0.425i)11-s + (−0.0506 − 0.998i)12-s + (0.820 − 0.571i)13-s + (−0.0168 + 0.999i)14-s + (−0.283 + 0.959i)15-s + (−0.972 + 0.234i)16-s + (0.918 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.365186297 - 0.4949076076i\)
\(L(\frac12)\) \(\approx\) \(2.365186297 - 0.4949076076i\)
\(L(1)\) \(\approx\) \(1.832719883 - 0.3984376844i\)
\(L(1)\) \(\approx\) \(1.832719883 - 0.3984376844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.664 - 0.747i)T \)
3 \( 1 + (0.997 + 0.0675i)T \)
5 \( 1 + (-0.217 + 0.975i)T \)
7 \( 1 + (-0.758 + 0.651i)T \)
11 \( 1 + (0.905 + 0.425i)T \)
13 \( 1 + (0.820 - 0.571i)T \)
17 \( 1 + (0.918 + 0.394i)T \)
19 \( 1 + (-0.688 + 0.724i)T \)
23 \( 1 + (0.440 - 0.897i)T \)
29 \( 1 + (0.470 - 0.882i)T \)
31 \( 1 + (-0.0506 + 0.998i)T \)
37 \( 1 + (0.638 - 0.769i)T \)
41 \( 1 + (-0.250 - 0.968i)T \)
43 \( 1 + (0.117 + 0.993i)T \)
47 \( 1 + (-0.857 + 0.514i)T \)
53 \( 1 + (-0.990 + 0.134i)T \)
59 \( 1 + (-0.999 + 0.0337i)T \)
61 \( 1 + (-0.997 - 0.0675i)T \)
67 \( 1 + (0.954 - 0.299i)T \)
71 \( 1 + (-0.801 - 0.598i)T \)
73 \( 1 + (-0.378 + 0.925i)T \)
79 \( 1 + (-0.585 - 0.810i)T \)
83 \( 1 + (0.990 - 0.134i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.918 - 0.394i)T \)
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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

−24.74655649027214561909060985883, −23.713379710508566174390543338427, −23.37924000220534377255039036562, −21.97221315592394233652545685925, −21.20642429863369430850104796740, −20.36146976482002518550255021120, −19.61484514797103851387031652735, −18.65468388695457122814813043862, −17.152562445757708065934736796181, −16.49857345531071271933549723097, −15.804754686183270227751933524301, −14.82241762943611402036254761088, −13.75237806733869406476517247186, −13.35886709145964975441689978618, −12.46936635011596428989454781604, −11.39488520277366884952023235761, −9.58709301691572089123403717864, −8.94198714596449038985137318894, −8.07183190466878688070054379688, −7.06563560402614454753471631006, −6.185284590026214603664706231605, −4.71849305114203827998005951401, −3.85611199140013265291503054149, −3.14499478273343270981220662856, −1.29421152408132715311450679377, 1.60036804784483215937877654503, 2.7817495938053074361294810591, 3.419076187771390773252714952622, 4.283655800731852826405206165982, 5.99225005464063630889995524227, 6.6766202563911934945666427620, 8.11985711504093577059623753552, 9.242560735744468432519518025119, 10.08767328570057221559284784512, 10.84973737279982211817187536866, 12.21614333868713308488025413949, 12.76033969481930866807456147765, 13.94252022141250665143243721830, 14.65209389885444595553697380213, 15.2172947890180183143044053653, 16.154060713300997312466983714932, 17.96737857002992323525154312093, 18.95936903533263863758777264699, 19.20737704893614053399802865781, 20.16208481404986976596024411590, 21.14404940066622578175615062501, 21.81240600027837605253307445912, 22.832325807299207559378152886956, 23.21334047986229392689217922273, 24.79136606613688346426923398362

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