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Properties

Label 2-819-91.89-c1-0-19
Degree $2$
Conductor $819$
Sign $-0.632 - 0.774i$
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  + (1.56 + 1.56i)2-s + 2.91i·4-s + (0.784 + 2.92i)5-s + (2.25 − 1.38i)7-s + (−1.43 + 1.43i)8-s + (−3.36 + 5.82i)10-s + (0.188 + 0.705i)11-s + (−2.65 + 2.44i)13-s + (5.70 + 1.35i)14-s + 1.33·16-s − 3.24·17-s + (3.85 + 1.03i)19-s + (−8.53 + 2.28i)20-s + (−0.809 + 1.40i)22-s − 5.96i·23-s + ⋯
L(s)  = 1  + (1.10 + 1.10i)2-s + 1.45i·4-s + (0.351 + 1.31i)5-s + (0.851 − 0.524i)7-s + (−0.506 + 0.506i)8-s + (−1.06 + 1.84i)10-s + (0.0569 + 0.212i)11-s + (−0.735 + 0.678i)13-s + (1.52 + 0.362i)14-s + 0.334·16-s − 0.787·17-s + (0.885 + 0.237i)19-s + (−1.90 + 0.511i)20-s + (−0.172 + 0.298i)22-s − 1.24i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33662 + 2.81865i\)
\(L(\frac12)\) \(\approx\) \(1.33662 + 2.81865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 + 1.38i)T \)
13 \( 1 + (2.65 - 2.44i)T \)
good2 \( 1 + (-1.56 - 1.56i)T + 2iT^{2} \)
5 \( 1 + (-0.784 - 2.92i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.188 - 0.705i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 + (-3.85 - 1.03i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.96iT - 23T^{2} \)
29 \( 1 + (2.78 + 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.99 - 1.07i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.97 - 6.97i)T - 37iT^{2} \)
41 \( 1 + (2.46 + 0.660i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.73 + 3.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.69 + 0.454i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.37 + 5.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.33 - 2.33i)T + 59iT^{2} \)
61 \( 1 + (-6.30 + 3.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.27 - 1.68i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.06 + 2.16i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.62 + 6.08i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.87 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 + (-6.62 - 6.62i)T + 89iT^{2} \)
97 \( 1 + (-2.37 - 8.86i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.50690525221838614771936318189, −9.886802013954141704719341570713, −8.454777024693468631153563069363, −7.54628185158426536265565911753, −6.83074723832263458105713357057, −6.42937405666361543236591255679, −5.16201507819686146693352772277, −4.48941711817159202516937160798, −3.42559231527673147293681611370, −2.14360309321955967871629988168, 1.23081425918086994913580321540, 2.20024162773645382725597477348, 3.41756912895859176287826393471, 4.65662677327913363217878598300, 5.18082208844329635806811773862, 5.72376731700962370099824414292, 7.42456494133549155040848567073, 8.445603265085381302642985925925, 9.234442318939856367061761721934, 10.08071821428911053551650827838

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