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Properties

Label 2-819-91.9-c1-0-14
Degree $2$
Conductor $819$
Sign $0.765 + 0.642i$
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  + (−1.19 − 2.06i)2-s + (−1.85 + 3.20i)4-s + (0.491 − 0.850i)5-s + (2.60 + 0.452i)7-s + 4.06·8-s − 2.34·10-s + 0.587·11-s + (2.39 + 2.69i)13-s + (−2.17 − 5.93i)14-s + (−1.15 − 1.99i)16-s + (−3.22 + 5.58i)17-s − 3.82·19-s + (1.81 + 3.14i)20-s + (−0.701 − 1.21i)22-s + (4.13 + 7.15i)23-s + ⋯
L(s)  = 1  + (−0.844 − 1.46i)2-s + (−0.925 + 1.60i)4-s + (0.219 − 0.380i)5-s + (0.985 + 0.170i)7-s + 1.43·8-s − 0.741·10-s + 0.177·11-s + (0.663 + 0.748i)13-s + (−0.581 − 1.58i)14-s + (−0.288 − 0.498i)16-s + (−0.782 + 1.35i)17-s − 0.877·19-s + (0.406 + 0.704i)20-s + (−0.149 − 0.259i)22-s + (0.861 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.765 + 0.642i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.765 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950093 - 0.345907i\)
\(L(\frac12)\) \(\approx\) \(0.950093 - 0.345907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.452i)T \)
13 \( 1 + (-2.39 - 2.69i)T \)
good2 \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.491 + 0.850i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.587T + 11T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 + (-4.13 - 7.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.98 - 3.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.49 - 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.877 + 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.83 + 3.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.19 + 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.17 - 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.212 - 0.368i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.00 + 5.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.20T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 + (-1.80 - 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.46 + 4.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + (1.04 + 1.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.84 + 6.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31013062947326581593962414044, −9.162625570674982314950936890993, −8.817819863561905837150189621254, −8.119512239655084612146463384977, −6.87321695557891163635400331045, −5.54883189443979704619228353322, −4.36319131062407535859785724283, −3.49493613706792918456301341016, −1.96143548006667330244874365771, −1.41371951953599393901834508263, 0.73973647940906414518716556145, 2.56004962916609093735604749358, 4.43382368284605920137642798783, 5.22378500233758032600989541538, 6.35537990927496952702720540402, 6.82969235610851416365819577557, 7.86872226099965835191149500765, 8.448510424141116833844529034629, 9.139020209638770385213061277333, 10.17907333292906194207981625490

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