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Properties

Label 2-819-91.76-c1-0-30
Degree $2$
Conductor $819$
Sign $0.464 + 0.885i$
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  + (0.281 − 1.04i)2-s + (0.710 + 0.410i)4-s + (1.02 − 1.02i)5-s + (2.23 + 1.41i)7-s + (2.16 − 2.16i)8-s + (−0.788 − 1.36i)10-s + (−0.972 − 0.260i)11-s + (−3.05 − 1.91i)13-s + (2.11 − 1.94i)14-s + (−0.842 − 1.45i)16-s + (2.37 − 4.10i)17-s + (0.391 + 1.46i)19-s + (1.15 − 0.308i)20-s + (−0.546 + 0.946i)22-s + (0.337 − 0.194i)23-s + ⋯
L(s)  = 1  + (0.198 − 0.741i)2-s + (0.355 + 0.205i)4-s + (0.459 − 0.459i)5-s + (0.843 + 0.536i)7-s + (0.765 − 0.765i)8-s + (−0.249 − 0.432i)10-s + (−0.293 − 0.0785i)11-s + (−0.847 − 0.530i)13-s + (0.565 − 0.519i)14-s + (−0.210 − 0.364i)16-s + (0.574 − 0.995i)17-s + (0.0899 + 0.335i)19-s + (0.257 − 0.0690i)20-s + (−0.116 + 0.201i)22-s + (0.0702 − 0.0405i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.464 + 0.885i$
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: \((2,\ 819,\ (\ :1/2),\ 0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01539 - 1.21832i\)
\(L(\frac12)\) \(\approx\) \(2.01539 - 1.21832i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 1.41i)T \)
13 \( 1 + (3.05 + 1.91i)T \)
good2 \( 1 + (-0.281 + 1.04i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.02 + 1.02i)T - 5iT^{2} \)
11 \( 1 + (0.972 + 0.260i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.37 + 4.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.391 - 1.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.337 + 0.194i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.30 - 7.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.92 + 2.92i)T - 31iT^{2} \)
37 \( 1 + (-9.77 - 2.61i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.64 + 2.31i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.28 + 4.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.78 + 4.78i)T + 47iT^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (0.889 - 0.238i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.36 - 4.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.56 + 0.687i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.46 + 2.46i)T + 73iT^{2} \)
79 \( 1 - 7.06T + 79T^{2} \)
83 \( 1 + (2.43 - 2.43i)T - 83iT^{2} \)
89 \( 1 + (2.79 - 10.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.55 - 13.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.13352373391087824302564813654, −9.508402699191795438149782977308, −8.332244790049989266416939787075, −7.65921020163023852502723428364, −6.65483573047963071061866107343, −5.27873870878616236065094279301, −4.85693557200273752073983553186, −3.31756544870798058342280176314, −2.41834348085177606200865951777, −1.30166649201670704504626910018, 1.60178860101003038067762845533, 2.68999218415822877585442926290, 4.39786184331759399391481027049, 5.09285424020047099848013596300, 6.23176719264120881684191785690, 6.75450313224073124384731433675, 7.83763633848190341495175758441, 8.207817604127986207910413137024, 9.831749123476402752492328617201, 10.24551307333412605712505052371

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