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Properties

Label 1-19e2-361.248-r0-0-0
Degree $1$
Conductor $361$
Sign $0.361 + 0.932i$
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  + (0.789 − 0.614i)2-s + (−0.677 + 0.735i)3-s + (0.245 − 0.969i)4-s + (−0.879 − 0.475i)5-s + (−0.0825 + 0.996i)6-s + (0.245 + 0.969i)7-s + (−0.401 − 0.915i)8-s + (−0.0825 − 0.996i)9-s + (−0.986 + 0.164i)10-s + (−0.0825 + 0.996i)11-s + (0.546 + 0.837i)12-s + (−0.677 + 0.735i)13-s + (0.789 + 0.614i)14-s + (0.945 − 0.324i)15-s + (−0.879 − 0.475i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.614i)2-s + (−0.677 + 0.735i)3-s + (0.245 − 0.969i)4-s + (−0.879 − 0.475i)5-s + (−0.0825 + 0.996i)6-s + (0.245 + 0.969i)7-s + (−0.401 − 0.915i)8-s + (−0.0825 − 0.996i)9-s + (−0.986 + 0.164i)10-s + (−0.0825 + 0.996i)11-s + (0.546 + 0.837i)12-s + (−0.677 + 0.735i)13-s + (0.789 + 0.614i)14-s + (0.945 − 0.324i)15-s + (−0.879 − 0.475i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7946117052 + 0.5441722801i\)
\(L(\frac12)\) \(\approx\) \(0.7946117052 + 0.5441722801i\)
\(L(1)\) \(\approx\) \(0.9959637051 + 0.04325340009i\)
\(L(1)\) \(\approx\) \(0.9959637051 + 0.04325340009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.789 - 0.614i)T \)
3 \( 1 + (-0.677 + 0.735i)T \)
5 \( 1 + (-0.879 - 0.475i)T \)
7 \( 1 + (0.245 + 0.969i)T \)
11 \( 1 + (-0.0825 + 0.996i)T \)
13 \( 1 + (-0.677 + 0.735i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
23 \( 1 + (-0.677 - 0.735i)T \)
29 \( 1 + (0.245 + 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.0825 + 0.996i)T \)
41 \( 1 + (0.945 - 0.324i)T \)
43 \( 1 + (-0.986 - 0.164i)T \)
47 \( 1 + (-0.0825 + 0.996i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (-0.401 + 0.915i)T \)
67 \( 1 + (-0.401 - 0.915i)T \)
71 \( 1 + (-0.401 - 0.915i)T \)
73 \( 1 + (0.245 + 0.969i)T \)
79 \( 1 + (-0.986 - 0.164i)T \)
83 \( 1 + (-0.879 + 0.475i)T \)
89 \( 1 + (0.245 - 0.969i)T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−24.358496094370604633214361503, −23.59847690918006346045013161361, −22.95397937246930986285703553229, −22.39684815502288828735760955263, −21.32088018135224928789813139449, −20.087858879722321653302995668466, −19.30625225550369797179790815088, −18.13131479445814670027400159828, −17.34533052792041220818467333502, −16.42757879673628653254309470107, −15.76345297059050537339148875509, −14.5296984917592605432732014225, −13.75183310088622495533065082001, −12.983106613552489659303550576746, −11.73113863458030586775990744864, −11.43503032110745375695460122150, −10.2494807779258117541058424455, −8.15197205160028114861798126180, −7.65744027729410691143721440582, −6.897846082590108266370519279713, −5.84860667688481056645379330297, −4.82935865741918904032036945170, −3.72430160361945053283673862158, −2.59718124969878096472607689527, −0.497264398035179934618081547287, 1.57922865461245905968030675992, 3.02005353755827535798787737353, 4.37022341857367510629997536743, 4.7105681329228410228697745359, 5.80610353111640918494508888003, 6.91800697645226563359162541109, 8.53609701584058711993023715988, 9.57391965452818807294310753819, 10.48515857524209582172940296081, 11.53483710277818448607823735469, 12.28091372576286876279109555363, 12.541885792267133977107443845308, 14.39632257350076774563597520963, 15.07812516673219333879902620111, 15.72302313413670254721893780771, 16.65466038454676144910132606786, 17.86573097666396287013119550769, 18.92479661564520490634384549736, 19.845370603997232907339383537325, 20.69948222825176482092555374978, 21.48741146121321574586401657269, 22.19937311175958969034348944430, 22.99780532890157477713899217748, 23.84469710499279948161523243352, 24.40429818174911927726697117606

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