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Properties

Label 2-819-91.80-c1-0-20
Degree $2$
Conductor $819$
Sign $-0.109 + 0.993i$
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  + (−0.568 − 2.12i)2-s + (−2.44 + 1.40i)4-s + (0.248 − 0.926i)5-s + (1.82 + 1.91i)7-s + (1.27 + 1.27i)8-s − 2.10·10-s + (2.00 + 2.00i)11-s + (2.97 − 2.03i)13-s + (3.03 − 4.95i)14-s + (−0.844 + 1.46i)16-s + (2.53 + 4.39i)17-s + (0.314 + 0.314i)19-s + (0.699 + 2.61i)20-s + (3.10 − 5.37i)22-s + (2.92 + 1.68i)23-s + ⋯
L(s)  = 1  + (−0.401 − 1.49i)2-s + (−1.22 + 0.704i)4-s + (0.111 − 0.414i)5-s + (0.688 + 0.724i)7-s + (0.449 + 0.449i)8-s − 0.665·10-s + (0.603 + 0.603i)11-s + (0.825 − 0.564i)13-s + (0.810 − 1.32i)14-s + (−0.211 + 0.365i)16-s + (0.615 + 1.06i)17-s + (0.0720 + 0.0720i)19-s + (0.156 + 0.584i)20-s + (0.662 − 1.14i)22-s + (0.609 + 0.351i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.109 + 0.993i$
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: \((2,\ 819,\ (\ :1/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943211 - 1.05304i\)
\(L(\frac12)\) \(\approx\) \(0.943211 - 1.05304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.91i)T \)
13 \( 1 + (-2.97 + 2.03i)T \)
good2 \( 1 + (0.568 + 2.12i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.248 + 0.926i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.00 - 2.00i)T + 11iT^{2} \)
17 \( 1 + (-2.53 - 4.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.314 - 0.314i)T + 19iT^{2} \)
23 \( 1 + (-2.92 - 1.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.91 + 8.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.21 + 0.861i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.09 + 1.36i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.13 - 4.21i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.57 + 1.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.10 + 2.43i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.30 + 7.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.83 - 0.758i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + (7.12 - 7.12i)T - 67iT^{2} \)
71 \( 1 + (-0.206 - 0.770i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.01 + 3.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.44 - 4.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.86 - 2.86i)T + 83iT^{2} \)
89 \( 1 + (-2.70 - 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.64 + 1.24i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.979705326166494218610234400149, −9.419098066944841394460026658919, −8.525122359289856212164029959557, −7.968801810144265027139273339214, −6.40204450942811804452836870831, −5.38228754767863009774976440938, −4.25516864850265480138029789383, −3.30234446460723476317493612001, −2.02682428712680601614126300311, −1.17033787747230925911726921258, 1.09938536791185375245749980799, 3.14088865926868364742320725454, 4.47953477007718989672360469471, 5.37652461397065749626796093186, 6.41705737526538138441208808008, 7.00033317809345599052754243020, 7.72827391605977920936608630474, 8.687019480380354305014869761184, 9.181316523003841581020307339119, 10.34006627390467752159688607042

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