Properties

Label 1-373-373.113-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.530 + 0.847i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.299 + 0.954i)2-s + (0.994 − 0.101i)3-s + (−0.820 + 0.571i)4-s + (0.897 + 0.440i)5-s + (0.394 + 0.918i)6-s + (0.874 − 0.485i)7-s + (−0.790 − 0.612i)8-s + (0.979 − 0.201i)9-s + (−0.151 + 0.988i)10-s + (−0.790 + 0.612i)11-s + (−0.758 + 0.651i)12-s + (0.612 + 0.790i)13-s + (0.724 + 0.688i)14-s + (0.937 + 0.347i)15-s + (0.347 − 0.937i)16-s + (−0.820 + 0.571i)17-s + ⋯
L(s)  = 1  + (0.299 + 0.954i)2-s + (0.994 − 0.101i)3-s + (−0.820 + 0.571i)4-s + (0.897 + 0.440i)5-s + (0.394 + 0.918i)6-s + (0.874 − 0.485i)7-s + (−0.790 − 0.612i)8-s + (0.979 − 0.201i)9-s + (−0.151 + 0.988i)10-s + (−0.790 + 0.612i)11-s + (−0.758 + 0.651i)12-s + (0.612 + 0.790i)13-s + (0.724 + 0.688i)14-s + (0.937 + 0.347i)15-s + (0.347 − 0.937i)16-s + (−0.820 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.530 + 0.847i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857960884 + 3.354675960i\)
\(L(\frac12)\) \(\approx\) \(1.857960884 + 3.354675960i\)
\(L(1)\) \(\approx\) \(1.567224073 + 1.194456786i\)
\(L(1)\) \(\approx\) \(1.567224073 + 1.194456786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.299 + 0.954i)T \)
3 \( 1 + (0.994 - 0.101i)T \)
5 \( 1 + (0.897 + 0.440i)T \)
7 \( 1 + (0.874 - 0.485i)T \)
11 \( 1 + (-0.790 + 0.612i)T \)
13 \( 1 + (0.612 + 0.790i)T \)
17 \( 1 + (-0.820 + 0.571i)T \)
19 \( 1 + (-0.937 + 0.347i)T \)
23 \( 1 + (0.101 - 0.994i)T \)
29 \( 1 + (-0.0506 + 0.998i)T \)
31 \( 1 + (0.758 + 0.651i)T \)
37 \( 1 + (0.250 + 0.968i)T \)
41 \( 1 + (0.918 - 0.394i)T \)
43 \( 1 + (0.571 + 0.820i)T \)
47 \( 1 + (-0.724 + 0.688i)T \)
53 \( 1 + (0.201 - 0.979i)T \)
59 \( 1 + (0.0506 - 0.998i)T \)
61 \( 1 + (0.101 + 0.994i)T \)
67 \( 1 + (0.897 + 0.440i)T \)
71 \( 1 + (-0.820 - 0.571i)T \)
73 \( 1 + (0.979 + 0.201i)T \)
79 \( 1 + (-0.988 - 0.151i)T \)
83 \( 1 + (0.979 + 0.201i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.571 - 0.820i)T \)
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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

−24.29511000799796859085758371902, −23.157286612097615456932309960545, −21.82224558830929383975367074167, −21.269104676027057468981179111532, −20.79272532199008833068790233621, −19.97421543382270833395066458594, −18.9268797735465803650822177294, −18.12674311994275595172216217517, −17.44270524754747686824800134280, −15.72082966682439677046151267776, −15.01974990506663256178327741267, −13.85726408397893283550434732715, −13.4162717076067096106653015123, −12.668843233261304855690821574843, −11.27813905647520505857513849951, −10.48828712551665550603369049356, −9.42411178405826396716312645288, −8.70327490915608552850153103799, −7.94477622377082143127593931128, −5.98194270115233952163738739365, −5.09230213999313554189024177336, −4.09873789340658385823193338012, −2.66813589819284175336898723502, −2.152820857821470773363110268662, −0.86531020027055957595091554993, 1.57366850452944916121027587776, 2.67910061443515883386630806180, 4.10910919324110038682524177317, 4.85181478173239415726734771798, 6.363738475497275424051636159883, 7.012483169441088205221216089, 8.157827292931443111236854510187, 8.774920988869335305027828949977, 9.930656788829258135282950637801, 10.86360244359422967743140267231, 12.68463950764070201006389798774, 13.2864040475673477638659462988, 14.262217274897078422369267780441, 14.60501019101486340819956834785, 15.5657167377028596129294833524, 16.67461370365368597485421941889, 17.77144749181768273793195091708, 18.20966217600304016290968136967, 19.24048882640050023317343623552, 20.80215894292298231569511905443, 21.04969165312806128010835677700, 22.01818733888730569446743307249, 23.21936351799981694895843088070, 24.00883203966490553672064706259, 24.70841274487989899680608685252

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