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Properties

Label 2-819-91.31-c1-0-30
Degree $2$
Conductor $819$
Sign $-0.0888 + 0.996i$
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  + (−0.694 − 0.186i)2-s + (−1.28 − 0.741i)4-s + (0.501 − 1.87i)5-s + (2.52 − 0.783i)7-s + (1.77 + 1.77i)8-s + (−0.696 + 1.20i)10-s + (3.08 − 0.825i)11-s + (−0.846 − 3.50i)13-s + (−1.90 + 0.0737i)14-s + (0.582 + 1.00i)16-s + (−0.254 + 0.440i)17-s + (−0.710 + 2.65i)19-s + (−2.03 + 2.03i)20-s − 2.29·22-s + (2.49 − 1.44i)23-s + ⋯
L(s)  = 1  + (−0.491 − 0.131i)2-s + (−0.642 − 0.370i)4-s + (0.224 − 0.836i)5-s + (0.955 − 0.296i)7-s + (0.626 + 0.626i)8-s + (−0.220 + 0.381i)10-s + (0.929 − 0.248i)11-s + (−0.234 − 0.972i)13-s + (−0.508 + 0.0196i)14-s + (0.145 + 0.252i)16-s + (−0.0616 + 0.106i)17-s + (−0.162 + 0.608i)19-s + (−0.454 + 0.454i)20-s − 0.488·22-s + (0.520 − 0.300i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0888 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0888 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0888 + 0.996i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0888 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797242 - 0.871560i\)
\(L(\frac12)\) \(\approx\) \(0.797242 - 0.871560i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 + 0.783i)T \)
13 \( 1 + (0.846 + 3.50i)T \)
good2 \( 1 + (0.694 + 0.186i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.501 + 1.87i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.08 + 0.825i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.254 - 0.440i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.710 - 2.65i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.49 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 + (-3.08 + 0.827i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.51 - 9.40i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.34 + 5.34i)T + 41iT^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 + (10.7 + 2.88i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.42 + 5.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.00 + 3.74i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.75 + 6.56i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.90 + 1.90i)T - 71iT^{2} \)
73 \( 1 + (-0.0676 - 0.252i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.86 - 5.86i)T + 83iT^{2} \)
89 \( 1 + (-11.8 - 3.17i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.04 - 7.04i)T + 97iT^{2} \)
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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

−10.09260138054681665694609521740, −8.976494032359758227473857800787, −8.524980289521686947177930726769, −7.82806142372654761711126412919, −6.51713326017810481237999772377, −5.23420148373148399572992433843, −4.87000160177628824473132710458, −3.67628481614819038227221824211, −1.76157959143873240575960529448, −0.812294593837914943318220096552, 1.48610991193862906173634116599, 2.90898704546835302269908745110, 4.25411651145662379815064676818, 4.90250086084547156790374327001, 6.39868534874871662649233694340, 7.08329413746568154125574031573, 7.958800758714247275103205383207, 8.910304662195591450175936919005, 9.390544352234919998562675402261, 10.34922907022044304511712115893

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