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Properties

Label 1-373-373.58-r1-0-0
Degree $1$
Conductor $373$
Sign $0.181 + 0.983i$
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  + (0.101 − 0.994i)2-s + (−0.528 + 0.848i)3-s + (−0.979 − 0.201i)4-s + (−0.988 + 0.151i)5-s + (0.790 + 0.612i)6-s + (−0.347 − 0.937i)7-s + (−0.299 + 0.954i)8-s + (−0.440 − 0.897i)9-s + (0.0506 + 0.998i)10-s + (−0.299 − 0.954i)11-s + (0.688 − 0.724i)12-s + (0.954 − 0.299i)13-s + (−0.968 + 0.250i)14-s + (0.394 − 0.918i)15-s + (0.918 + 0.394i)16-s + (−0.979 − 0.201i)17-s + ⋯
L(s)  = 1  + (0.101 − 0.994i)2-s + (−0.528 + 0.848i)3-s + (−0.979 − 0.201i)4-s + (−0.988 + 0.151i)5-s + (0.790 + 0.612i)6-s + (−0.347 − 0.937i)7-s + (−0.299 + 0.954i)8-s + (−0.440 − 0.897i)9-s + (0.0506 + 0.998i)10-s + (−0.299 − 0.954i)11-s + (0.688 − 0.724i)12-s + (0.954 − 0.299i)13-s + (−0.968 + 0.250i)14-s + (0.394 − 0.918i)15-s + (0.918 + 0.394i)16-s + (−0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1061792047 - 0.08837893864i\)
\(L(\frac12)\) \(\approx\) \(-0.1061792047 - 0.08837893864i\)
\(L(1)\) \(\approx\) \(0.4720552158 - 0.3009731593i\)
\(L(1)\) \(\approx\) \(0.4720552158 - 0.3009731593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.101 - 0.994i)T \)
3 \( 1 + (-0.528 + 0.848i)T \)
5 \( 1 + (-0.988 + 0.151i)T \)
7 \( 1 + (-0.347 - 0.937i)T \)
11 \( 1 + (-0.299 - 0.954i)T \)
13 \( 1 + (0.954 - 0.299i)T \)
17 \( 1 + (-0.979 - 0.201i)T \)
19 \( 1 + (-0.394 - 0.918i)T \)
23 \( 1 + (0.848 - 0.528i)T \)
29 \( 1 + (-0.874 - 0.485i)T \)
31 \( 1 + (-0.688 - 0.724i)T \)
37 \( 1 + (-0.820 - 0.571i)T \)
41 \( 1 + (-0.612 + 0.790i)T \)
43 \( 1 + (0.201 - 0.979i)T \)
47 \( 1 + (0.968 + 0.250i)T \)
53 \( 1 + (-0.897 - 0.440i)T \)
59 \( 1 + (0.874 + 0.485i)T \)
61 \( 1 + (0.848 + 0.528i)T \)
67 \( 1 + (-0.988 + 0.151i)T \)
71 \( 1 + (-0.979 + 0.201i)T \)
73 \( 1 + (-0.440 + 0.897i)T \)
79 \( 1 + (0.998 - 0.0506i)T \)
83 \( 1 + (-0.440 + 0.897i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.201 + 0.979i)T \)
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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

−25.16570747842637725364060222999, −23.99209238654609645332427513821, −23.540414291707831185572522092481, −22.730669227872375803965326162914, −22.12315465445651681889884524237, −20.71720342775591649345367691951, −19.37661883276758047292619027687, −18.71772803601799953247098180032, −18.053224795086948173804951614398, −17.070393718790537351170733599895, −16.09439314660327833288038147942, −15.47777076945364461579442241287, −14.58505660980427562715717397668, −13.215538658145731724202994623578, −12.66076112015741131852309164663, −11.86315359228329845585270827796, −10.71302767375369241280640009610, −9.06417355292284857784921314069, −8.39237741755592844272294020545, −7.35982022399052572749533589666, −6.64542105132309827730020610577, −5.63189434661667950787533030469, −4.69353070811946041812675261054, −3.420898143041006623826280023425, −1.66501310102579784111394425626, 0.06470359709901740839934721092, 0.721469230007702659575477496205, 2.90341963242523224064048065460, 3.78969726649330089544966484079, 4.38152332628398554675525483473, 5.597184100074693978284851304491, 6.90066010791737282928902665138, 8.42174512288984990896489422427, 9.16046479013465975546485182917, 10.51621228544701477229928010835, 10.99533191969632944214809165964, 11.48768751773926368767276239905, 12.86805730330561903209270247266, 13.577833907193475188906951759222, 14.84706647766595729061876294828, 15.72800846964177505503348817052, 16.60837198324931064069168849017, 17.55438580328470602340956811991, 18.647931389693554282641895988003, 19.47140688568490165507488977325, 20.4278136529157764093195124507, 20.866168491042701229579479742912, 22.166620368697288555143792777320, 22.56402838051198786114150314725, 23.55168293751519151200469489364

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