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Properties

Label 2-819-91.9-c1-0-41
Degree $2$
Conductor $819$
Sign $-0.835 + 0.549i$
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  + (−0.425 − 0.737i)2-s + (0.637 − 1.10i)4-s + (1.72 − 2.98i)5-s + (1.82 − 1.91i)7-s − 2.78·8-s − 2.92·10-s + 0.897·11-s + (−3.07 − 1.88i)13-s + (−2.18 − 0.525i)14-s + (−0.0891 − 0.154i)16-s + (0.968 − 1.67i)17-s + 1.03·19-s + (−2.19 − 3.80i)20-s + (−0.382 − 0.661i)22-s + (2.82 + 4.89i)23-s + ⋯
L(s)  = 1  + (−0.300 − 0.521i)2-s + (0.318 − 0.552i)4-s + (0.769 − 1.33i)5-s + (0.688 − 0.725i)7-s − 0.985·8-s − 0.926·10-s + 0.270·11-s + (−0.852 − 0.522i)13-s + (−0.585 − 0.140i)14-s + (−0.0222 − 0.0386i)16-s + (0.234 − 0.406i)17-s + 0.238·19-s + (−0.490 − 0.850i)20-s + (−0.0814 − 0.141i)22-s + (0.589 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473698 - 1.58395i\)
\(L(\frac12)\) \(\approx\) \(0.473698 - 1.58395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.91i)T \)
13 \( 1 + (3.07 + 1.88i)T \)
good2 \( 1 + (0.425 + 0.737i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.897T + 11T^{2} \)
17 \( 1 + (-0.968 + 1.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 1.03T + 19T^{2} \)
23 \( 1 + (-2.82 - 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.917 - 1.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.56 - 7.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.66 - 4.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.95 - 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.59 + 6.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.69 + 8.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.255 - 0.442i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.43T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + (1.72 + 2.98i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.45 + 9.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.253 + 0.438i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.888063377504483632318302543681, −9.291953311468789521469825573564, −8.411198156778745000692991455828, −7.40240954583542255821365245119, −6.30104657959340141683889268218, −5.13942197848740333581653809537, −4.85376606210590011129079559848, −3.09758570609559509010598359666, −1.66477981080553606107440480977, −0.946698641515663881175875243017, 2.23631522575548826548757992495, 2.76238145817294933161918181266, 4.22599135184795675336974224864, 5.70843895583587046060392440479, 6.30460727452862128319528305009, 7.19746605245672248735355653809, 7.81639781160970655681544223463, 8.905390347928818620169564042830, 9.565468814175141929435856349389, 10.60403262243201995361385652290

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