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Properties

Label 2-819-91.74-c1-0-15
Degree $2$
Conductor $819$
Sign $0.992 + 0.122i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.268·2-s − 1.92·4-s + (−1.28 − 2.21i)5-s + (1.80 + 1.93i)7-s − 1.05·8-s + (−0.343 − 0.594i)10-s + (1.97 + 3.41i)11-s + (−3.15 + 1.74i)13-s + (0.483 + 0.518i)14-s + 3.57·16-s − 0.785·17-s + (3.74 − 6.49i)19-s + (2.46 + 4.27i)20-s + (0.529 + 0.916i)22-s + 7.95·23-s + ⋯
L(s)  = 1  + 0.189·2-s − 0.964·4-s + (−0.572 − 0.992i)5-s + (0.681 + 0.731i)7-s − 0.372·8-s + (−0.108 − 0.188i)10-s + (0.594 + 1.03i)11-s + (−0.874 + 0.484i)13-s + (0.129 + 0.138i)14-s + 0.893·16-s − 0.190·17-s + (0.859 − 1.48i)19-s + (0.552 + 0.956i)20-s + (0.112 + 0.195i)22-s + 1.65·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29930 - 0.0796270i\)
\(L(\frac12)\) \(\approx\) \(1.29930 - 0.0796270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.80 - 1.93i)T \)
13 \( 1 + (3.15 - 1.74i)T \)
good2 \( 1 - 0.268T + 2T^{2} \)
5 \( 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.97 - 3.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.785T + 17T^{2} \)
19 \( 1 + (-3.74 + 6.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.95T + 23T^{2} \)
29 \( 1 + (-1.17 + 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.27 + 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 + (1.21 - 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.63 + 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.96T + 59T^{2} \)
61 \( 1 + (4.72 - 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.15 - 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.384 - 0.665i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + 7.66T + 89T^{2} \)
97 \( 1 + (-1.18 - 2.05i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.869233581818648369351231329997, −9.180027127256012729964083012243, −8.768406017521125739347510129612, −7.79000254550402976585093473012, −6.87401392633016836081341085530, −5.34511778733123592656400128311, −4.74321818184931505334761510211, −4.28491751512050240978706909365, −2.63222610059935418708890490608, −0.959570804471549353224845178751, 0.945389770868697198052926512062, 3.10057409972799192504338787962, 3.74069489761454791152321944988, 4.80405593509564093649767837692, 5.71811266425802233510536606104, 6.95342885804616772756266883278, 7.70253270924784391156111120538, 8.446045064083301191659034831341, 9.439063046342151194967087477032, 10.37055328989275377723364419861

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