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Properties

Label 2-819-91.33-c1-0-13
Degree $2$
Conductor $819$
Sign $-0.845 - 0.534i$
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  + (−0.170 + 0.636i)2-s + (1.35 + 0.782i)4-s + (1.02 + 3.81i)5-s + (−2.47 − 0.938i)7-s + (−1.66 + 1.66i)8-s − 2.60·10-s + (1.90 − 1.90i)11-s + (−0.360 + 3.58i)13-s + (1.01 − 1.41i)14-s + (0.791 + 1.37i)16-s + (−1.67 + 2.89i)17-s + (4.69 − 4.69i)19-s + (−1.60 + 5.98i)20-s + (0.886 + 1.53i)22-s + (−5.65 + 3.26i)23-s + ⋯
L(s)  = 1  + (−0.120 + 0.450i)2-s + (0.678 + 0.391i)4-s + (0.457 + 1.70i)5-s + (−0.935 − 0.354i)7-s + (−0.587 + 0.587i)8-s − 0.823·10-s + (0.573 − 0.573i)11-s + (−0.0999 + 0.994i)13-s + (0.272 − 0.378i)14-s + (0.197 + 0.342i)16-s + (−0.405 + 0.702i)17-s + (1.07 − 1.07i)19-s + (−0.358 + 1.33i)20-s + (0.189 + 0.327i)22-s + (−1.17 + 0.681i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.845 - 0.534i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.423681 + 1.46372i\)
\(L(\frac12)\) \(\approx\) \(0.423681 + 1.46372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.47 + 0.938i)T \)
13 \( 1 + (0.360 - 3.58i)T \)
good2 \( 1 + (0.170 - 0.636i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.02 - 3.81i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.90 + 1.90i)T - 11iT^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.69 + 4.69i)T - 19iT^{2} \)
23 \( 1 + (5.65 - 3.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.95 + 3.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.41 - 0.378i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.02 + 0.542i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.717 - 2.67i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.41 + 5.43i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.76 - 0.473i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.90 + 6.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.574 + 0.153i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 5.48iT - 61T^{2} \)
67 \( 1 + (2.33 + 2.33i)T + 67iT^{2} \)
71 \( 1 + (-0.629 + 2.35i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.670 - 2.50i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.485 + 0.485i)T - 83iT^{2} \)
89 \( 1 + (-2.12 + 7.93i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.37 - 1.97i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.58093991178535278237313672429, −9.783305745620073751555974322304, −8.958773441636952555759009557493, −7.65407794451871845031315175802, −6.98849880161155629124879207494, −6.41151034390096741253137935228, −5.89185482475111067084344500302, −3.88946274398943778071169898679, −3.10564884541706442728161717873, −2.16458436308903660047552828685, 0.75284812571492343106215419786, 1.90886699029037692560914839970, 3.14813055279910525605686910581, 4.49827688352023270405369390546, 5.61736007122729560687129430068, 6.10625693659890075268262677340, 7.31727550023400374201144231620, 8.402175611116133408962484057319, 9.409500027376574997067441362722, 9.705545203600110547663949693878

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