Properties

Label 1-373-373.264-r0-0-0
Degree $1$
Conductor $373$
Sign $0.492 - 0.870i$
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  + (−0.347 + 0.937i)2-s + (0.918 − 0.394i)3-s + (−0.758 − 0.651i)4-s + (0.250 − 0.968i)5-s + (0.0506 + 0.998i)6-s + (−0.440 − 0.897i)7-s + (0.874 − 0.485i)8-s + (0.688 − 0.724i)9-s + (0.820 + 0.571i)10-s + (0.874 + 0.485i)11-s + (−0.954 − 0.299i)12-s + (−0.874 − 0.485i)13-s + (0.994 − 0.101i)14-s + (−0.151 − 0.988i)15-s + (0.151 + 0.988i)16-s + (−0.758 − 0.651i)17-s + ⋯
L(s)  = 1  + (−0.347 + 0.937i)2-s + (0.918 − 0.394i)3-s + (−0.758 − 0.651i)4-s + (0.250 − 0.968i)5-s + (0.0506 + 0.998i)6-s + (−0.440 − 0.897i)7-s + (0.874 − 0.485i)8-s + (0.688 − 0.724i)9-s + (0.820 + 0.571i)10-s + (0.874 + 0.485i)11-s + (−0.954 − 0.299i)12-s + (−0.874 − 0.485i)13-s + (0.994 − 0.101i)14-s + (−0.151 − 0.988i)15-s + (0.151 + 0.988i)16-s + (−0.758 − 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077537504 - 0.6286123105i\)
\(L(\frac12)\) \(\approx\) \(1.077537504 - 0.6286123105i\)
\(L(1)\) \(\approx\) \(1.074734377 - 0.1301596203i\)
\(L(1)\) \(\approx\) \(1.074734377 - 0.1301596203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.347 + 0.937i)T \)
3 \( 1 + (0.918 - 0.394i)T \)
5 \( 1 + (0.250 - 0.968i)T \)
7 \( 1 + (-0.440 - 0.897i)T \)
11 \( 1 + (0.874 + 0.485i)T \)
13 \( 1 + (-0.874 - 0.485i)T \)
17 \( 1 + (-0.758 - 0.651i)T \)
19 \( 1 + (-0.151 + 0.988i)T \)
23 \( 1 + (-0.918 - 0.394i)T \)
29 \( 1 + (0.979 + 0.201i)T \)
31 \( 1 + (-0.954 + 0.299i)T \)
37 \( 1 + (0.528 - 0.848i)T \)
41 \( 1 + (-0.0506 - 0.998i)T \)
43 \( 1 + (0.758 + 0.651i)T \)
47 \( 1 + (0.994 + 0.101i)T \)
53 \( 1 + (-0.688 - 0.724i)T \)
59 \( 1 + (0.979 + 0.201i)T \)
61 \( 1 + (-0.918 + 0.394i)T \)
67 \( 1 + (0.250 - 0.968i)T \)
71 \( 1 + (-0.758 + 0.651i)T \)
73 \( 1 + (0.688 + 0.724i)T \)
79 \( 1 + (-0.820 - 0.571i)T \)
83 \( 1 + (0.688 + 0.724i)T \)
89 \( 1 + T \)
97 \( 1 + (0.758 + 0.651i)T \)
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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

−25.1644877830547908864424882248, −23.94175224152364945260907145009, −22.306238085987235882866707863, −21.81793062184295155036407858941, −21.64585711215389607328110398069, −20.12494127793305887266806732143, −19.445174871414778394533396019365, −18.97447414495929675248818109058, −18.00370242825704186377500098878, −16.99835479027715817330788077866, −15.73117007950872637854941072172, −14.78461669087748107080932511757, −13.995705869256927961459398842316, −13.14395021437892992483391328000, −11.975391022376301272239456923787, −11.088721789445701164092739831540, −10.05342435457314532912392014616, −9.3195601311776773242690745974, −8.66619213559689511978441224714, −7.43636940186453566941991308833, −6.23678323376646276087560587461, −4.579836647165919349982058818674, −3.552301448470906341308114641485, −2.62472214112128042459646562900, −1.96395226093102889601823120345, 0.76293757515032071861987749519, 2.00077710424086681343656551972, 3.87811108066883483943782924019, 4.62632820715075549856579112017, 6.06625660188256809211481384956, 7.08529403341901502312518673913, 7.79070432102921845523013843534, 8.837388396885821155581779449035, 9.545020832275467923957109484884, 10.28050619883061500456931987045, 12.32042757077031357560339473457, 12.95177065975477980556073025183, 14.04193226435398037625225936444, 14.4252010547598876436250824352, 15.71122849353129400417559160756, 16.46031014288360645160831367075, 17.391752060403436324972867287335, 18.04118432321390490635147770180, 19.40704958894347587659901584378, 19.91001827170257289763623847087, 20.5559418866730806416308850590, 22.028750448263827534162961241581, 22.99642554672442946717532675484, 23.96666036875922660169286461807, 24.64678090150653590654945189568

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