Properties

Label 1-373-373.204-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: $\chi_{373} (204, \cdot )$
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

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

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