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Properties

Label 2-819-91.80-c1-0-4
Degree $2$
Conductor $819$
Sign $-0.452 + 0.891i$
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  + (0.604 + 2.25i)2-s + (−2.99 + 1.73i)4-s + (−0.721 + 2.69i)5-s + (−1.83 + 1.90i)7-s + (−2.41 − 2.41i)8-s − 6.51·10-s + (4.28 + 4.28i)11-s + (−0.711 − 3.53i)13-s + (−5.40 − 3.00i)14-s + (0.530 − 0.918i)16-s + (−0.641 − 1.11i)17-s + (−1.37 − 1.37i)19-s + (−2.49 − 9.32i)20-s + (−7.08 + 12.2i)22-s + (1.85 + 1.07i)23-s + ⋯
L(s)  = 1  + (0.427 + 1.59i)2-s + (−1.49 + 0.865i)4-s + (−0.322 + 1.20i)5-s + (−0.695 + 0.718i)7-s + (−0.854 − 0.854i)8-s − 2.06·10-s + (1.29 + 1.29i)11-s + (−0.197 − 0.980i)13-s + (−1.44 − 0.802i)14-s + (0.132 − 0.229i)16-s + (−0.155 − 0.269i)17-s + (−0.314 − 0.314i)19-s + (−0.558 − 2.08i)20-s + (−1.50 + 2.61i)22-s + (0.387 + 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.452 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721301 - 1.17426i\)
\(L(\frac12)\) \(\approx\) \(0.721301 - 1.17426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.90i)T \)
13 \( 1 + (0.711 + 3.53i)T \)
good2 \( 1 + (-0.604 - 2.25i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.721 - 2.69i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.28 - 4.28i)T + 11iT^{2} \)
17 \( 1 + (0.641 + 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.37 + 1.37i)T + 19iT^{2} \)
23 \( 1 + (-1.85 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.48 + 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.92 - 0.515i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-11.4 + 3.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.19 - 4.44i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.94 + 1.70i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.99 - 2.67i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.30 - 7.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.2 + 3.00i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 3.67iT - 61T^{2} \)
67 \( 1 + (-1.20 + 1.20i)T - 67iT^{2} \)
71 \( 1 + (-1.73 - 6.47i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.17 + 4.37i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.418 - 0.724i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.01 - 2.01i)T + 83iT^{2} \)
89 \( 1 + (1.29 + 4.81i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.27 - 1.68i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.75391640348023068674394781737, −9.602472348898170578305292184219, −9.058003462150012128710942860824, −7.76887973725802621308446695186, −7.24532111411899243350312531340, −6.47321597818285780934229161166, −5.93086719564803371119006629366, −4.72450003593415330455223724536, −3.75168170593846118517341510897, −2.56870132698212873946756769357, 0.62439984563264693771516720093, 1.59762463069092194882745547279, 3.23033468390926794297276380175, 4.03186169899705875818479788876, 4.58110189993378946218819871557, 5.90927753689570640149269011655, 6.96762979538083308621187914951, 8.429026129433597498822771591053, 9.161238247970201458947346942693, 9.642699994135629112145109287838

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