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Properties

Label 1-373-373.38-r0-0-0
Degree $1$
Conductor $373$
Sign $0.930 - 0.367i$
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  + (0.890 + 0.455i)2-s + (0.857 − 0.514i)3-s + (0.585 + 0.810i)4-s + (−0.184 − 0.982i)5-s + (0.997 − 0.0675i)6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (0.470 − 0.882i)9-s + (0.283 − 0.959i)10-s + (−0.931 + 0.363i)11-s + (0.918 + 0.394i)12-s + (0.151 − 0.988i)13-s + (0.990 − 0.134i)14-s + (−0.664 − 0.747i)15-s + (−0.315 + 0.948i)16-s + (−0.994 + 0.101i)17-s + ⋯
L(s)  = 1  + (0.890 + 0.455i)2-s + (0.857 − 0.514i)3-s + (0.585 + 0.810i)4-s + (−0.184 − 0.982i)5-s + (0.997 − 0.0675i)6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (0.470 − 0.882i)9-s + (0.283 − 0.959i)10-s + (−0.931 + 0.363i)11-s + (0.918 + 0.394i)12-s + (0.151 − 0.988i)13-s + (0.990 − 0.134i)14-s + (−0.664 − 0.747i)15-s + (−0.315 + 0.948i)16-s + (−0.994 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.930 - 0.367i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.930 - 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.839852119 - 0.5405893598i\)
\(L(\frac12)\) \(\approx\) \(2.839852119 - 0.5405893598i\)
\(L(1)\) \(\approx\) \(2.191828730 - 0.1557283538i\)
\(L(1)\) \(\approx\) \(2.191828730 - 0.1557283538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.890 + 0.455i)T \)
3 \( 1 + (0.857 - 0.514i)T \)
5 \( 1 + (-0.184 - 0.982i)T \)
7 \( 1 + (0.820 - 0.571i)T \)
11 \( 1 + (-0.931 + 0.363i)T \)
13 \( 1 + (0.151 - 0.988i)T \)
17 \( 1 + (-0.994 + 0.101i)T \)
19 \( 1 + (0.979 + 0.201i)T \)
23 \( 1 + (-0.874 + 0.485i)T \)
29 \( 1 + (-0.713 + 0.701i)T \)
31 \( 1 + (0.918 - 0.394i)T \)
37 \( 1 + (0.736 + 0.676i)T \)
41 \( 1 + (-0.440 + 0.897i)T \)
43 \( 1 + (0.585 + 0.810i)T \)
47 \( 1 + (-0.378 - 0.925i)T \)
53 \( 1 + (0.470 + 0.882i)T \)
59 \( 1 + (0.963 + 0.266i)T \)
61 \( 1 + (0.857 - 0.514i)T \)
67 \( 1 + (-0.758 + 0.651i)T \)
71 \( 1 + (0.409 + 0.912i)T \)
73 \( 1 + (-0.999 - 0.0337i)T \)
79 \( 1 + (0.283 - 0.959i)T \)
83 \( 1 + (0.470 + 0.882i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.994 + 0.101i)T \)
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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

−24.360819896341880367079944539379, −23.96744141056860336379577637756, −22.55316463769652405969468224269, −21.99329600332174164266523909878, −21.169385729180911221900632057422, −20.60716650576726359270432531366, −19.48979510415771246322863427732, −18.77340905381856739509643495042, −18.00510843380573462914937369162, −16.0291965295924116450526362841, −15.585301891242938231358440690867, −14.66889324750394244442259633229, −14.01745399634829912856274346520, −13.34894313578882434544924354677, −11.857912487291843688433948728312, −11.13822988811534732870567181303, −10.35565122061054491159465058604, −9.28328669562225719141596280429, −8.06424495835046418458618714151, −7.045657794466835148779543270174, −5.76416890007858459791218064855, −4.65263194554685718791711325627, −3.77265199565255289642442802078, −2.58493357315283316354033233277, −2.0860769177126267920775239913, 1.381613264093751127697491641259, 2.60434059221884242508512878539, 3.84786539214367156704210475738, 4.74698942699340658983774454586, 5.71614413952009324430513055721, 7.18607049827174346227543928649, 7.963614541080337351531988883775, 8.39455410040370011769627224865, 9.88329109744852715016122610567, 11.32813764599673340261129607538, 12.278765341992656454623053340241, 13.33710186136963820695786381797, 13.44872696191618885932004996910, 14.76424486912931787631609757132, 15.45714045758327268168249374368, 16.30552673252601400460887546164, 17.58565169080570498783486781170, 18.08449712569747198260464856357, 19.83125136374628437084033018364, 20.41129581574259357258474852273, 20.75471624755759927717734674324, 21.89824163758001823386678801701, 23.22032629237418810536166535273, 23.81293718044095895475836952300, 24.51656578145917801783502803057

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