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Properties

Label 2-819-91.88-c1-0-22
Degree $2$
Conductor $819$
Sign $-0.536 + 0.844i$
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  + (−1.99 − 1.15i)2-s + (1.65 + 2.86i)4-s + (0.733 − 0.423i)5-s + (2.09 − 1.62i)7-s − 3.00i·8-s − 1.95·10-s + 1.50i·11-s + (−2.92 − 2.11i)13-s + (−6.03 + 0.824i)14-s + (−0.156 + 0.271i)16-s + (−1.03 − 1.79i)17-s + 0.0474i·19-s + (2.42 + 1.40i)20-s + (1.73 − 2.99i)22-s + (3.90 − 6.77i)23-s + ⋯
L(s)  = 1  + (−1.41 − 0.814i)2-s + (0.826 + 1.43i)4-s + (0.328 − 0.189i)5-s + (0.790 − 0.612i)7-s − 1.06i·8-s − 0.617·10-s + 0.453i·11-s + (−0.810 − 0.585i)13-s + (−1.61 + 0.220i)14-s + (−0.0391 + 0.0678i)16-s + (−0.251 − 0.435i)17-s + 0.0108i·19-s + (0.542 + 0.313i)20-s + (0.369 − 0.639i)22-s + (0.815 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.536 + 0.844i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362018 - 0.658815i\)
\(L(\frac12)\) \(\approx\) \(0.362018 - 0.658815i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 1.62i)T \)
13 \( 1 + (2.92 + 2.11i)T \)
good2 \( 1 + (1.99 + 1.15i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.733 + 0.423i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.50iT - 11T^{2} \)
17 \( 1 + (1.03 + 1.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.0474iT - 19T^{2} \)
23 \( 1 + (-3.90 + 6.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.679 - 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.80 - 3.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.80 + 3.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.67 + 5.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.63 + 8.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.311 - 0.180i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.35 + 2.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.42 - 0.820i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 - 2.04iT - 67T^{2} \)
71 \( 1 + (12.3 + 7.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.85 + 3.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (15.1 + 8.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.369 - 0.213i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.15641235497883311399918684775, −9.099096587409745477922028221066, −8.554715526968313139240845511777, −7.52410570528670815334184942548, −7.04878738474968962200507011239, −5.39008323808799873117718848714, −4.43959316847131996424951241272, −2.89389606710056641510017585846, −1.89887382194733207715495828704, −0.64623729498030285129165465571, 1.34799654392159031365236696099, 2.58530434119430906418874575511, 4.43703961835278956098592604351, 5.64924172697378596226369460307, 6.33272211312485930679242231545, 7.33965280862642362064309030932, 8.034984378913527159374218542684, 8.775532489509644985866398774420, 9.518414029687752039193900736216, 10.14967245695497551616120450309

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