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Properties

Label 1-373-373.39-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.660 - 0.750i$
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  + (0.736 − 0.676i)2-s + (0.585 − 0.810i)3-s + (0.0843 − 0.996i)4-s + (0.997 + 0.0675i)5-s + (−0.117 − 0.993i)6-s + (−0.874 − 0.485i)7-s + (−0.612 − 0.790i)8-s + (−0.315 − 0.948i)9-s + (0.780 − 0.625i)10-s + (0.990 + 0.134i)11-s + (−0.758 − 0.651i)12-s + (−0.612 + 0.790i)13-s + (−0.972 + 0.234i)14-s + (0.638 − 0.769i)15-s + (−0.985 − 0.168i)16-s + (0.820 + 0.571i)17-s + ⋯
L(s)  = 1  + (0.736 − 0.676i)2-s + (0.585 − 0.810i)3-s + (0.0843 − 0.996i)4-s + (0.997 + 0.0675i)5-s + (−0.117 − 0.993i)6-s + (−0.874 − 0.485i)7-s + (−0.612 − 0.790i)8-s + (−0.315 − 0.948i)9-s + (0.780 − 0.625i)10-s + (0.990 + 0.134i)11-s + (−0.758 − 0.651i)12-s + (−0.612 + 0.790i)13-s + (−0.972 + 0.234i)14-s + (0.638 − 0.769i)15-s + (−0.985 − 0.168i)16-s + (0.820 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014808501 - 2.245454043i\)
\(L(\frac12)\) \(\approx\) \(1.014808501 - 2.245454043i\)
\(L(1)\) \(\approx\) \(1.382485860 - 1.322070504i\)
\(L(1)\) \(\approx\) \(1.382485860 - 1.322070504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.736 - 0.676i)T \)
3 \( 1 + (0.585 - 0.810i)T \)
5 \( 1 + (0.997 + 0.0675i)T \)
7 \( 1 + (-0.874 - 0.485i)T \)
11 \( 1 + (0.990 + 0.134i)T \)
13 \( 1 + (-0.612 + 0.790i)T \)
17 \( 1 + (0.820 + 0.571i)T \)
19 \( 1 + (0.347 - 0.937i)T \)
23 \( 1 + (-0.994 + 0.101i)T \)
29 \( 1 + (-0.839 + 0.543i)T \)
31 \( 1 + (-0.758 + 0.651i)T \)
37 \( 1 + (0.963 - 0.266i)T \)
41 \( 1 + (0.918 + 0.394i)T \)
43 \( 1 + (0.0843 - 0.996i)T \)
47 \( 1 + (0.283 + 0.959i)T \)
53 \( 1 + (-0.315 + 0.948i)T \)
59 \( 1 + (0.890 + 0.455i)T \)
61 \( 1 + (0.585 - 0.810i)T \)
67 \( 1 + (-0.440 - 0.897i)T \)
71 \( 1 + (-0.905 - 0.425i)T \)
73 \( 1 + (-0.664 - 0.747i)T \)
79 \( 1 + (0.780 - 0.625i)T \)
83 \( 1 + (-0.315 + 0.948i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.820 + 0.571i)T \)
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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

−25.11138005972445696897120257918, −24.429723692070592021907965148798, −22.735639007887589023269415447795, −22.32939055738278683764731592266, −21.70690943176328320309987678695, −20.74209387122354016245737736860, −20.04196415874069998666453655186, −18.759430445425267941780167017554, −17.56073778044682088245977805636, −16.51519918566649131654623389084, −16.20694181976071947248701456917, −14.820616868606525937978161325443, −14.489879097566905297989148968272, −13.47561992723758350205991965566, −12.660792985251833994819394996209, −11.58590577672826337748250010524, −9.929830386052793616688236067575, −9.53370429291445461504730498760, −8.42594493166348323495480831006, −7.31730612526976354114454101188, −5.88357239942323108180723000508, −5.55938945765843428207967892166, −4.13530209646318864513391478944, −3.19778682998327871174411747577, −2.26662898393617713417349276799, 1.188698091474634579355037405024, 2.12048055081594127433850754743, 3.167942848127031774915397340584, 4.156146653002510213265113524518, 5.72616316966835539265108358502, 6.51506707247562115611578922794, 7.28926151366228485940287974562, 9.3457321310896102070076644822, 9.39986359960871294714711257510, 10.72137971978852930493559431365, 12.0136451457532741070555292848, 12.69255507540599370216491242054, 13.5240941034334789125807319636, 14.23186821848301393626541595327, 14.76689107930537389166242747018, 16.33448252115277036749598227524, 17.386568467170783228174532651092, 18.3960405930737092229952410130, 19.3467816765537490115055438863, 19.84214693177063882073949996201, 20.70043107982937890173333215251, 21.85208105956557891086219194680, 22.28024337170062937936542965502, 23.54275039773432047267008440297, 24.12327187226866718043108994114

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