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Properties

Label 1-373-373.24-r1-0-0
Degree $1$
Conductor $373$
Sign $0.183 - 0.983i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (−0.598 − 0.801i)2-s + (0.378 + 0.925i)3-s + (−0.283 + 0.959i)4-s + (−0.996 + 0.0843i)5-s + (0.514 − 0.857i)6-s + (−0.151 + 0.988i)7-s + (0.937 − 0.347i)8-s + (−0.713 + 0.701i)9-s + (0.664 + 0.747i)10-s + (−0.168 − 0.985i)11-s + (−0.994 + 0.101i)12-s + (−0.347 + 0.937i)13-s + (0.882 − 0.470i)14-s + (−0.455 − 0.890i)15-s + (−0.839 − 0.543i)16-s + (−0.688 − 0.724i)17-s + ⋯
L(s)  = 1  + (−0.598 − 0.801i)2-s + (0.378 + 0.925i)3-s + (−0.283 + 0.959i)4-s + (−0.996 + 0.0843i)5-s + (0.514 − 0.857i)6-s + (−0.151 + 0.988i)7-s + (0.937 − 0.347i)8-s + (−0.713 + 0.701i)9-s + (0.664 + 0.747i)10-s + (−0.168 − 0.985i)11-s + (−0.994 + 0.101i)12-s + (−0.347 + 0.937i)13-s + (0.882 − 0.470i)14-s + (−0.455 − 0.890i)15-s + (−0.839 − 0.543i)16-s + (−0.688 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.183 - 0.983i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.183 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2364343430 - 0.1963818225i\)
\(L(\frac12)\) \(\approx\) \(0.2364343430 - 0.1963818225i\)
\(L(1)\) \(\approx\) \(0.5643146354 + 0.06701495235i\)
\(L(1)\) \(\approx\) \(0.5643146354 + 0.06701495235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.598 - 0.801i)T \)
3 \( 1 + (0.378 + 0.925i)T \)
5 \( 1 + (-0.996 + 0.0843i)T \)
7 \( 1 + (-0.151 + 0.988i)T \)
11 \( 1 + (-0.168 - 0.985i)T \)
13 \( 1 + (-0.347 + 0.937i)T \)
17 \( 1 + (-0.688 - 0.724i)T \)
19 \( 1 + (-0.998 + 0.0506i)T \)
23 \( 1 + (-0.790 + 0.612i)T \)
29 \( 1 + (0.997 - 0.0675i)T \)
31 \( 1 + (0.994 + 0.101i)T \)
37 \( 1 + (-0.943 - 0.331i)T \)
41 \( 1 + (-0.874 + 0.485i)T \)
43 \( 1 + (-0.959 - 0.283i)T \)
47 \( 1 + (0.0337 + 0.999i)T \)
53 \( 1 + (0.701 - 0.713i)T \)
59 \( 1 + (0.557 + 0.830i)T \)
61 \( 1 + (0.925 - 0.378i)T \)
67 \( 1 + (0.571 + 0.820i)T \)
71 \( 1 + (0.972 + 0.234i)T \)
73 \( 1 + (0.963 - 0.266i)T \)
79 \( 1 + (0.747 - 0.664i)T \)
83 \( 1 + (-0.713 - 0.701i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.724 + 0.688i)T \)
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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

−24.56046942106521602855771012908, −23.78776728865356941151446602216, −23.21867552390546987723260447393, −22.56550057542700726020027475069, −20.49396203966290727155963253634, −19.85031153668886957662373944725, −19.42675729565555952301253953367, −18.33656671460142916009707500981, −17.47656371420680792304278973545, −16.875604175098318824536590935778, −15.49405015014541920368679915103, −15.03626044816494688265226852619, −13.969973043199828243097042511699, −12.99625848056390200236264210989, −12.14915383360230294000497978085, −10.73497610375365165183619316626, −9.979195644985384991663137435353, −8.33967119000831810070705487594, −8.171026415585016453392751542081, −6.98111994678450333146012189158, −6.58875813952647309268666801394, −4.91065809776707855707937167041, −3.84526053140483844290267656979, −2.171362965175463947530573950585, −0.767245973247165241010077605789, 0.14278730702920417221343431738, 2.22118593426689353787337122017, 3.103700604484142502932970819263, 4.05752592686179588409741606539, 4.999018149141376681695516526678, 6.70723694755127160875260719370, 8.25614062432150971688346595399, 8.58274563726144679605930152278, 9.54732722007141945596490056381, 10.5731065775723280472246015589, 11.56591231326715361172557730849, 11.93927736959443272634246454822, 13.37601752184928312911876606191, 14.40392111270192290721631020717, 15.68684620379507290136674605313, 16.0308641366473296230369692260, 17.0582106968706319617927189619, 18.343136189751049647949699990656, 19.3174355985954292095741758970, 19.51665394966641576097927104857, 20.73752321719726872699313618013, 21.54483006508003407980137162582, 22.0372829108686551428208858570, 23.02098940418021194682122510531, 24.308349029768911965168401724076

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