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Properties

Label 2-819-91.76-c1-0-2
Degree $2$
Conductor $819$
Sign $-0.411 + 0.911i$
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  + (−0.385 + 1.43i)2-s + (−0.189 − 0.109i)4-s + (−1.07 + 1.07i)5-s + (−0.612 + 2.57i)7-s + (−1.87 + 1.87i)8-s + (−1.12 − 1.95i)10-s + (−1.68 − 0.451i)11-s + (−3.51 − 0.818i)13-s + (−3.46 − 1.87i)14-s + (−2.19 − 3.80i)16-s + (1.43 − 2.48i)17-s + (−0.389 − 1.45i)19-s + (0.319 − 0.0856i)20-s + (1.29 − 2.25i)22-s + (3.21 − 1.85i)23-s + ⋯
L(s)  = 1  + (−0.272 + 1.01i)2-s + (−0.0945 − 0.0545i)4-s + (−0.479 + 0.479i)5-s + (−0.231 + 0.972i)7-s + (−0.663 + 0.663i)8-s + (−0.357 − 0.618i)10-s + (−0.508 − 0.136i)11-s + (−0.973 − 0.227i)13-s + (−0.926 − 0.500i)14-s + (−0.548 − 0.950i)16-s + (0.348 − 0.602i)17-s + (−0.0893 − 0.333i)19-s + (0.0714 − 0.0191i)20-s + (0.277 − 0.479i)22-s + (0.669 − 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.411 + 0.911i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263113 - 0.407514i\)
\(L(\frac12)\) \(\approx\) \(0.263113 - 0.407514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.612 - 2.57i)T \)
13 \( 1 + (3.51 + 0.818i)T \)
good2 \( 1 + (0.385 - 1.43i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.07 - 1.07i)T - 5iT^{2} \)
11 \( 1 + (1.68 + 0.451i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.389 + 1.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.21 + 1.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.65 - 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.32 - 1.32i)T - 31iT^{2} \)
37 \( 1 + (5.83 + 1.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.10 + 0.830i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.29 - 1.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.86 + 5.86i)T + 47iT^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (-0.967 + 0.259i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.0305 - 0.0176i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 - 4.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (11.4 - 3.05i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 + (10.8 - 10.8i)T - 83iT^{2} \)
89 \( 1 + (-2.02 + 7.54i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.46 - 9.21i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.82194487251939503093470640459, −9.739956954051524154763900781950, −8.867623743527988680532020468198, −8.178525481527085585715420305440, −7.21382684180164242659591185916, −6.81683171604986880423510177780, −5.57351314408015687239323840850, −5.03568210103992488544739512135, −3.22222088653426659632251519929, −2.49242451264694045015461200232, 0.24807474672344883973579319327, 1.61957793033821995794337175296, 2.94949816995955651783199971879, 3.94183473079810497056601422331, 4.87610421037256251068596706757, 6.22343092633648127616697225532, 7.21365238032689425098911918569, 7.994252143101807607752918263036, 9.064368472670993218916799983281, 9.990430677843980546814668994930

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