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Properties

Label 1-19e2-361.158-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.142 + 0.989i$
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  + (0.997 − 0.0734i)2-s + (−0.703 + 0.710i)3-s + (0.989 − 0.146i)4-s + (−0.227 + 0.973i)5-s + (−0.649 + 0.760i)6-s + (0.0275 + 0.999i)7-s + (0.975 − 0.218i)8-s + (−0.00918 − 0.999i)9-s + (−0.155 + 0.987i)10-s + (0.350 − 0.936i)11-s + (−0.592 + 0.805i)12-s + (0.577 + 0.816i)13-s + (0.100 + 0.994i)14-s + (−0.531 − 0.847i)15-s + (0.957 − 0.289i)16-s + (−0.621 + 0.783i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0734i)2-s + (−0.703 + 0.710i)3-s + (0.989 − 0.146i)4-s + (−0.227 + 0.973i)5-s + (−0.649 + 0.760i)6-s + (0.0275 + 0.999i)7-s + (0.975 − 0.218i)8-s + (−0.00918 − 0.999i)9-s + (−0.155 + 0.987i)10-s + (0.350 − 0.936i)11-s + (−0.592 + 0.805i)12-s + (0.577 + 0.816i)13-s + (0.100 + 0.994i)14-s + (−0.531 − 0.847i)15-s + (0.957 − 0.289i)16-s + (−0.621 + 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.142 + 0.989i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.142 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210113250 + 1.396297115i\)
\(L(\frac12)\) \(\approx\) \(1.210113250 + 1.396297115i\)
\(L(1)\) \(\approx\) \(1.363860119 + 0.6786522835i\)
\(L(1)\) \(\approx\) \(1.363860119 + 0.6786522835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.997 - 0.0734i)T \)
3 \( 1 + (-0.703 + 0.710i)T \)
5 \( 1 + (-0.227 + 0.973i)T \)
7 \( 1 + (0.0275 + 0.999i)T \)
11 \( 1 + (0.350 - 0.936i)T \)
13 \( 1 + (0.577 + 0.816i)T \)
17 \( 1 + (-0.621 + 0.783i)T \)
23 \( 1 + (-0.263 + 0.964i)T \)
29 \( 1 + (0.315 - 0.948i)T \)
31 \( 1 + (-0.962 + 0.272i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (0.209 + 0.977i)T \)
43 \( 1 + (-0.777 + 0.628i)T \)
47 \( 1 + (-0.333 - 0.942i)T \)
53 \( 1 + (0.983 + 0.182i)T \)
59 \( 1 + (0.741 - 0.670i)T \)
61 \( 1 + (0.888 - 0.459i)T \)
67 \( 1 + (0.606 - 0.794i)T \)
71 \( 1 + (0.888 + 0.459i)T \)
73 \( 1 + (-0.367 - 0.929i)T \)
79 \( 1 + (0.933 + 0.359i)T \)
83 \( 1 + (0.451 - 0.892i)T \)
89 \( 1 + (-0.367 + 0.929i)T \)
97 \( 1 + (0.606 + 0.794i)T \)
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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

−24.22130778641167187623035676597, −23.619212331107876173096916338871, −22.79647748705570360815844003686, −22.322022704313219914417222958, −20.792526781898149882471785636996, −20.2479490459490483536093539157, −19.591955294847271318734385868856, −18.04025887470051530332218922034, −17.2146628315545366208906545867, −16.41614959793447138716387104668, −15.715910156461897355088459770085, −14.36532609808087314786008712865, −13.41004419756663915112748566031, −12.79039792566343894527989249043, −12.09073746571995460651917076271, −11.10881270188156506680823489562, −10.2343464274048420385091967796, −8.49653798700128849016920805540, −7.366706595767310020043166518571, −6.791732719357628262070635695533, −5.495453797582128049234024451171, −4.736133637450316994252568085625, −3.80616254577311989999893848258, −2.07869817060388479467457273976, −0.92236734778741093344047965310, 1.9246996214681611634350811772, 3.32901900511254333656602423491, 3.9448219595516497671177130484, 5.27641064489631580437041691579, 6.18174452691603808368541969279, 6.67989977879362096240499584463, 8.35068511815500086169265096714, 9.65171832627307857649536694300, 10.82727561540536472838866699906, 11.43653212462792043678147874741, 11.96952476867661585726650607638, 13.32702016641516855822820299190, 14.37164658301891885104435440199, 15.17048898683208589779087546868, 15.78879741334254818555270320280, 16.62597725491670979319243302071, 17.8712296882244752859839974311, 18.900549215609157655949508057667, 19.71138977845054557906860166828, 21.176050118225259245525821344, 21.70481864805515005766999954468, 22.106435998880952309145730255940, 23.11899114190075533438748051420, 23.76473242689968601702862822946, 24.73410939521179043762261441544

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