↑

Properties

Label 1-373-373.108-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

Related objects

Downloads

Learn more

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: $\chi_{373} (108, \cdot )$
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 \( 1 + (-0.994 - 0.101i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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