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Properties

Label 2-819-91.76-c1-0-12
Degree $2$
Conductor $819$
Sign $0.842 - 0.538i$
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 + (−1.22 + 2.34i)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 + (−0.462 + 0.886i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.842 - 0.538i$
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: \((2,\ 819,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61050 + 0.470775i\)
\(L(\frac12)\) \(\approx\) \(1.61050 + 0.470775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 2.34i)T \)
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

−10.77482964230380502693538240561, −9.542327202915577882918979820678, −8.705523789579745046649999310355, −7.77508812702874319662582291567, −6.72211817391180435826049821038, −6.13483111211218730283536106371, −4.68531371344633510959448016915, −3.56120444989713398267433661946, −2.87930990359661510581400193350, −1.67810423529736384958655050387, 0.796974251458986168239413952997, 2.56947930493510314955660995668, 4.01364154954968558151734844523, 4.80670432513136588899806608788, 5.97835075848017926849430303888, 6.59275949396636560822653723056, 7.70734780883104827161746326213, 8.001519284321143339031508372048, 9.277446195297550290124460702347, 10.21895492635998157222070352308

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