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Properties

Label 1-373-373.64-r0-0-0
Degree $1$
Conductor $373$
Sign $0.639 - 0.768i$
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  + (0.994 + 0.101i)2-s + (0.528 − 0.848i)3-s + (0.979 + 0.201i)4-s + (−0.151 − 0.988i)5-s + (0.612 − 0.790i)6-s + (0.347 + 0.937i)7-s + (0.954 + 0.299i)8-s + (−0.440 − 0.897i)9-s + (−0.0506 − 0.998i)10-s + (0.954 − 0.299i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.250 + 0.968i)14-s + (−0.918 − 0.394i)15-s + (0.918 + 0.394i)16-s + (0.979 + 0.201i)17-s + ⋯
L(s)  = 1  + (0.994 + 0.101i)2-s + (0.528 − 0.848i)3-s + (0.979 + 0.201i)4-s + (−0.151 − 0.988i)5-s + (0.612 − 0.790i)6-s + (0.347 + 0.937i)7-s + (0.954 + 0.299i)8-s + (−0.440 − 0.897i)9-s + (−0.0506 − 0.998i)10-s + (0.954 − 0.299i)11-s + (0.688 − 0.724i)12-s + (−0.954 + 0.299i)13-s + (0.250 + 0.968i)14-s + (−0.918 − 0.394i)15-s + (0.918 + 0.394i)16-s + (0.979 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.639 - 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.635156572 - 1.234719711i\)
\(L(\frac12)\) \(\approx\) \(2.635156572 - 1.234719711i\)
\(L(1)\) \(\approx\) \(2.123368891 - 0.6071505307i\)
\(L(1)\) \(\approx\) \(2.123368891 - 0.6071505307i\)

Euler product

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

−24.75069630277865275126928881954, −23.58357100710534237894491416352, −22.84293565640705292624390554256, −22.055093830873854020834771074282, −21.47755139394664820330441313312, −20.38626146753822686348519857059, −19.78503918893287311768580770668, −19.040890449719146083268132807932, −17.31084001004949862664393048083, −16.65956839823449070569373455504, −15.40744092807165107159570920758, −14.73473313198740671549736599674, −14.270372713997794273423947719344, −13.41461079476244676970929857706, −12.008318586601529982765257530936, −11.12749388444042211013696197907, −10.338328625120859700661790828701, −9.59417877490884467518009834135, −7.781943444135697169603623252839, −7.21309018948922464171600044419, −5.94158688147205486622646274661, −4.66017733193677779611863043791, −3.90516574210356191920413224850, −3.08199216775304631015706072047, −1.91731126886079610411918210221, 1.460526972722754901854833596725, 2.32426821828501585366830738428, 3.624845847246871057204571787720, 4.73649592491242408402253605973, 5.818414251814328958490947960623, 6.65722743313140795978418545325, 7.9774587144110902435595790707, 8.524013305859732989086853140583, 9.75480124311824863763452800756, 11.607868395527714111341519050550, 12.13165405807019404425822067671, 12.66822261942461821194924506236, 13.739810952929393251923517063896, 14.65045340435109364305395044875, 15.14680920669802707805735036233, 16.6222255311726624446194953536, 17.07616452369550543887843004739, 18.574737412070364002007898565455, 19.463480570080235384485730352745, 20.12083464104078568532826457047, 21.118942433761187364682125762902, 21.7340890600909865756573590023, 22.92806020577998145897379677437, 23.84979541377054526778639568075, 24.525840309166293952666704196649

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