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Properties

Label 1-373-373.111-r0-0-0
Degree $1$
Conductor $373$
Sign $0.104 - 0.994i$
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  + (−0.440 + 0.897i)2-s + (0.151 − 0.988i)3-s + (−0.612 − 0.790i)4-s + (−0.994 − 0.101i)5-s + (0.820 + 0.571i)6-s + (0.688 − 0.724i)7-s + (0.979 − 0.201i)8-s + (−0.954 − 0.299i)9-s + (0.528 − 0.848i)10-s + (0.979 + 0.201i)11-s + (−0.874 + 0.485i)12-s + (0.979 + 0.201i)13-s + (0.347 + 0.937i)14-s + (−0.250 + 0.968i)15-s + (−0.250 + 0.968i)16-s + (−0.612 − 0.790i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.897i)2-s + (0.151 − 0.988i)3-s + (−0.612 − 0.790i)4-s + (−0.994 − 0.101i)5-s + (0.820 + 0.571i)6-s + (0.688 − 0.724i)7-s + (0.979 − 0.201i)8-s + (−0.954 − 0.299i)9-s + (0.528 − 0.848i)10-s + (0.979 + 0.201i)11-s + (−0.874 + 0.485i)12-s + (0.979 + 0.201i)13-s + (0.347 + 0.937i)14-s + (−0.250 + 0.968i)15-s + (−0.250 + 0.968i)16-s + (−0.612 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.104 - 0.994i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5996611508 - 0.5397316111i\)
\(L(\frac12)\) \(\approx\) \(0.5996611508 - 0.5397316111i\)
\(L(1)\) \(\approx\) \(0.7579886841 - 0.1530596225i\)
\(L(1)\) \(\approx\) \(0.7579886841 - 0.1530596225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.440 + 0.897i)T \)
3 \( 1 + (0.151 - 0.988i)T \)
5 \( 1 + (-0.994 - 0.101i)T \)
7 \( 1 + (0.688 - 0.724i)T \)
11 \( 1 + (0.979 + 0.201i)T \)
13 \( 1 + (0.979 + 0.201i)T \)
17 \( 1 + (-0.612 - 0.790i)T \)
19 \( 1 + (-0.250 - 0.968i)T \)
23 \( 1 + (0.151 + 0.988i)T \)
29 \( 1 + (-0.758 - 0.651i)T \)
31 \( 1 + (-0.874 - 0.485i)T \)
37 \( 1 + (0.918 - 0.394i)T \)
41 \( 1 + (0.820 + 0.571i)T \)
43 \( 1 + (-0.612 - 0.790i)T \)
47 \( 1 + (0.347 - 0.937i)T \)
53 \( 1 + (-0.954 + 0.299i)T \)
59 \( 1 + (-0.758 - 0.651i)T \)
61 \( 1 + (0.151 - 0.988i)T \)
67 \( 1 + (-0.994 - 0.101i)T \)
71 \( 1 + (-0.612 + 0.790i)T \)
73 \( 1 + (-0.954 + 0.299i)T \)
79 \( 1 + (0.528 - 0.848i)T \)
83 \( 1 + (-0.954 + 0.299i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.612 - 0.790i)T \)
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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

−25.07911660020521311906827224415, −23.76429983786926797549080145, −22.61613189923797949320171778336, −22.13021974203814626212564101033, −21.15345415023948727527009995611, −20.46228677037568173434692664882, −19.69207642098969803808155375221, −18.829601127215295158383965696800, −17.950663924521717054188812033372, −16.77124816914174852114588963053, −16.111388129953799342926634848243, −14.91084857506111003534816199296, −14.36786993390712582061383670705, −12.82280088898474652958096237816, −11.867933731845284761119244219029, −11.07368735429397042183555950412, −10.60592882438829856383771676722, −9.08765554573950869769218676753, −8.6530061274620946867744308588, −7.82952922843915187748161804910, −6.03658464391775217021428575303, −4.56063628055051301622563539656, −3.918135057367626795202812293991, −3.002227251429732145784086050154, −1.53847687661949828073029543177, 0.61526716650597968597561066380, 1.707126206326702365741457304699, 3.72413344477601318314391986858, 4.69146605002581024703189103239, 6.097980349068455856207087606793, 7.14665599111139074631311667668, 7.54227605212753533888092071103, 8.60175832471402494027773371795, 9.293235507864790912889427836053, 11.18761099743543528201187741490, 11.424587351854954727636665440241, 13.04053970936293576260407241023, 13.75815842638844100909846143046, 14.657666470274621994757393331243, 15.47506954121535734299336990340, 16.59952309985230660307698156318, 17.36270230679866748146843154859, 18.163552649315849017443852291586, 19.00095616360133076850719287573, 19.892382811635204261035956312744, 20.35619215733844672913887925463, 22.191423505385449661847441084076, 23.368234428096718884091543192733, 23.47451206997123561222869549673, 24.447872164613742168548155148832

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