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Properties

Label 1-373-373.16-r0-0-0
Degree $1$
Conductor $373$
Sign $0.887 - 0.461i$
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  + (0.997 + 0.0675i)2-s + (−0.931 − 0.363i)3-s + (0.990 + 0.134i)4-s + (0.409 − 0.912i)5-s + (−0.905 − 0.425i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.736 + 0.676i)9-s + (0.470 − 0.882i)10-s + (−0.664 − 0.747i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (0.638 + 0.769i)14-s + (−0.713 + 0.701i)15-s + (0.963 + 0.266i)16-s + (−0.612 + 0.790i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0675i)2-s + (−0.931 − 0.363i)3-s + (0.990 + 0.134i)4-s + (0.409 − 0.912i)5-s + (−0.905 − 0.425i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.736 + 0.676i)9-s + (0.470 − 0.882i)10-s + (−0.664 − 0.747i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (0.638 + 0.769i)14-s + (−0.713 + 0.701i)15-s + (0.963 + 0.266i)16-s + (−0.612 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.887 - 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.096527557 - 0.5125493428i\)
\(L(\frac12)\) \(\approx\) \(2.096527557 - 0.5125493428i\)
\(L(1)\) \(\approx\) \(1.676432884 - 0.2448737130i\)
\(L(1)\) \(\approx\) \(1.676432884 - 0.2448737130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.997 + 0.0675i)T \)
3 \( 1 + (-0.931 - 0.363i)T \)
5 \( 1 + (0.409 - 0.912i)T \)
7 \( 1 + (0.688 + 0.724i)T \)
11 \( 1 + (-0.664 - 0.747i)T \)
13 \( 1 + (0.979 - 0.201i)T \)
17 \( 1 + (-0.612 + 0.790i)T \)
19 \( 1 + (-0.250 + 0.968i)T \)
23 \( 1 + (0.151 - 0.988i)T \)
29 \( 1 + (0.943 + 0.331i)T \)
31 \( 1 + (-0.874 + 0.485i)T \)
37 \( 1 + (-0.117 - 0.993i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (0.990 + 0.134i)T \)
47 \( 1 + (-0.985 - 0.168i)T \)
53 \( 1 + (0.736 - 0.676i)T \)
59 \( 1 + (-0.184 - 0.982i)T \)
61 \( 1 + (-0.931 - 0.363i)T \)
67 \( 1 + (-0.994 + 0.101i)T \)
71 \( 1 + (-0.378 + 0.925i)T \)
73 \( 1 + (0.217 + 0.975i)T \)
79 \( 1 + (0.470 - 0.882i)T \)
83 \( 1 + (0.736 - 0.676i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.612 + 0.790i)T \)
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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

−24.26837256760950682542245312632, −23.48794415858673136864740962195, −22.97459380580166938256943044967, −22.19580858187918665938011147075, −21.2360368151248323753118957846, −20.82241374773051313142893572777, −19.64019554017665551501797089790, −18.15851438773612999609476972455, −17.71401862574207747711991036653, −16.583907909803176110519528182746, −15.52704116512181231309258892259, −15.02192897740223575524208873864, −13.7416916858121816561645213833, −13.260716472971474204270716684356, −11.86492970036858966548240933544, −11.02614931024630119340072274466, −10.68069893187597733350992048802, −9.5632075471342368845968023085, −7.53093228727081873431537681437, −6.85483407469975447220703524046, −5.921308498740156238035003729177, −4.86989415651928717336949224380, −4.13512611280622432432553783798, −2.81687317325503503260391064848, −1.48003330251628043228105591736, 1.31083200363004459007029958273, 2.28351511418273497814761040277, 4.02042613304434077530448718388, 5.06406142312021305159161219063, 5.743589661528053263536680238064, 6.36352367372494144579465382127, 7.93049488908223380905533610453, 8.657754173282383001006767467800, 10.562569870745031389914977484922, 11.050980396066340415244835092189, 12.27114259510392373630583883944, 12.713205785999949541224972640803, 13.566758279146836172916864877997, 14.60686465849084373469093539535, 15.99079385067919888239070055790, 16.217580639672536633489466368192, 17.41181065804684022736573380445, 18.20574685575846032122368089952, 19.29491182794072468193122025009, 20.62251232389769321356278907948, 21.30121301044507677172007264880, 21.79998573758171859918395936157, 22.96793622796252205029209814210, 23.683202756457228209752314648362, 24.45523312980745485505815836217

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