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Properties

Label 1-19e2-361.77-r0-0-0
Degree $1$
Conductor $361$
Sign $0.377 + 0.925i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.0825 − 0.996i)2-s + (−0.879 + 0.475i)3-s + (−0.986 + 0.164i)4-s + (0.945 − 0.324i)5-s + (0.546 + 0.837i)6-s + (−0.986 − 0.164i)7-s + (0.245 + 0.969i)8-s + (0.546 − 0.837i)9-s + (−0.401 − 0.915i)10-s + (0.546 + 0.837i)11-s + (0.789 − 0.614i)12-s + (−0.879 + 0.475i)13-s + (−0.0825 + 0.996i)14-s + (−0.677 + 0.735i)15-s + (0.945 − 0.324i)16-s + (−0.986 − 0.164i)17-s + ⋯
L(s)  = 1  + (−0.0825 − 0.996i)2-s + (−0.879 + 0.475i)3-s + (−0.986 + 0.164i)4-s + (0.945 − 0.324i)5-s + (0.546 + 0.837i)6-s + (−0.986 − 0.164i)7-s + (0.245 + 0.969i)8-s + (0.546 − 0.837i)9-s + (−0.401 − 0.915i)10-s + (0.546 + 0.837i)11-s + (0.789 − 0.614i)12-s + (−0.879 + 0.475i)13-s + (−0.0825 + 0.996i)14-s + (−0.677 + 0.735i)15-s + (0.945 − 0.324i)16-s + (−0.986 − 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.377 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3467231721 + 0.2330394039i\)
\(L(\frac12)\) \(\approx\) \(0.3467231721 + 0.2330394039i\)
\(L(1)\) \(\approx\) \(0.5976425903 - 0.1174052828i\)
\(L(1)\) \(\approx\) \(0.5976425903 - 0.1174052828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.0825 - 0.996i)T \)
3 \( 1 + (-0.879 + 0.475i)T \)
5 \( 1 + (0.945 - 0.324i)T \)
7 \( 1 + (-0.986 - 0.164i)T \)
11 \( 1 + (0.546 + 0.837i)T \)
13 \( 1 + (-0.879 + 0.475i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
23 \( 1 + (-0.879 - 0.475i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (-0.0825 + 0.996i)T \)
37 \( 1 + (0.546 + 0.837i)T \)
41 \( 1 + (-0.677 + 0.735i)T \)
43 \( 1 + (-0.401 + 0.915i)T \)
47 \( 1 + (0.546 + 0.837i)T \)
53 \( 1 + (0.546 + 0.837i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (0.245 - 0.969i)T \)
67 \( 1 + (0.245 + 0.969i)T \)
71 \( 1 + (0.245 + 0.969i)T \)
73 \( 1 + (-0.986 - 0.164i)T \)
79 \( 1 + (-0.401 + 0.915i)T \)
83 \( 1 + (0.945 + 0.324i)T \)
89 \( 1 + (-0.986 + 0.164i)T \)
97 \( 1 + (0.245 - 0.969i)T \)
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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

−24.56609655705588178634570144032, −23.91322018225277565714220549153, −22.68199537054867274546188802084, −22.12290063489692892373010590090, −21.79310454491534341594614395133, −19.80727671779706298275113542908, −18.88262809549570701931973950333, −18.1895914287542178710319189244, −17.25006108471914526478739708094, −16.786217797462856808533642072553, −15.830313299447193206043797163421, −14.78423561920766756794064442810, −13.55202598936080511363002382205, −13.19506050047595144974180420380, −12.067637006802832534826201829976, −10.681962610097063328073566740511, −9.80616055408149228417555875273, −8.939272260327765729393306515659, −7.50306799522130078037803107190, −6.6407885677841417147520750793, −5.952133401994753092478466775009, −5.31040710462981625655442253286, −3.78132069416873138064708912597, −2.09166063878917904638446007960, −0.29786186982473079812992660015, 1.43526368907141709976257784469, 2.65171025509317778694353547644, 4.14585673757661317603375515830, 4.81445140168246054157575835331, 6.03192591562505642958590960771, 6.982523894564274194660125496178, 8.918798846784681703061289576709, 9.75012594420631841069535356423, 10.03041931091732512257500070499, 11.23772605246127924832141313971, 12.2749431778618381447590693392, 12.806952615650152924207743433267, 13.84566901397515790944264381668, 14.970457474081143241151093493198, 16.380288708983259144248072201443, 17.05895377045190595940990417417, 17.75755028019690019954332845054, 18.620525048922225206383097300795, 19.917556430195289632920736275725, 20.408972059455661173855211575147, 21.66298457446761001842172678502, 22.078250254131761811559718130676, 22.70408942131145033552948180268, 23.73193719963688163058198860963, 24.887705199664477903402237310945

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