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Properties

Label 2-819-91.31-c1-0-41
Degree $2$
Conductor $819$
Sign $0.355 + 0.934i$
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  + (1.84 + 0.493i)2-s + (1.41 + 0.818i)4-s + (0.885 − 3.30i)5-s + (−2.39 − 1.12i)7-s + (−0.489 − 0.489i)8-s + (3.26 − 5.65i)10-s + (−1.66 + 0.445i)11-s + (3.57 + 0.501i)13-s + (−3.84 − 3.26i)14-s + (−2.29 − 3.97i)16-s + (−1.22 + 2.12i)17-s + (1.34 − 5.03i)19-s + (3.96 − 3.96i)20-s − 3.28·22-s + (3.97 − 2.29i)23-s + ⋯
L(s)  = 1  + (1.30 + 0.349i)2-s + (0.708 + 0.409i)4-s + (0.396 − 1.47i)5-s + (−0.904 − 0.426i)7-s + (−0.173 − 0.173i)8-s + (1.03 − 1.78i)10-s + (−0.501 + 0.134i)11-s + (0.990 + 0.138i)13-s + (−1.02 − 0.871i)14-s + (−0.574 − 0.994i)16-s + (−0.297 + 0.515i)17-s + (0.309 − 1.15i)19-s + (0.885 − 0.885i)20-s − 0.700·22-s + (0.828 − 0.478i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.355 + 0.934i$
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: \((2,\ 819,\ (\ :1/2),\ 0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18563 - 1.50637i\)
\(L(\frac12)\) \(\approx\) \(2.18563 - 1.50637i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 + 1.12i)T \)
13 \( 1 + (-3.57 - 0.501i)T \)
good2 \( 1 + (-1.84 - 0.493i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.885 + 3.30i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.66 - 0.445i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.34 + 5.03i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.97 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.184T + 29T^{2} \)
31 \( 1 + (-2.46 + 0.659i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.0563 + 0.210i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.63 - 4.63i)T + 41iT^{2} \)
43 \( 1 + 0.562iT - 43T^{2} \)
47 \( 1 + (3.72 + 0.998i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.67 + 4.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.73 - 13.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.30 - 0.754i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.78 - 6.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.70 - 1.70i)T - 71iT^{2} \)
73 \( 1 + (-3.15 - 11.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.48 + 2.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.504 - 0.504i)T + 83iT^{2} \)
89 \( 1 + (-7.20 - 1.92i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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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

−9.959563090436702248922259138833, −9.151894288761024643711733475854, −8.500328936690324805392854640279, −7.13998704675741043486902593265, −6.30776606663951387820699455631, −5.52262716956689420971302632693, −4.69593506205901766129205791812, −3.98142989183080706376883789533, −2.77502069393836613738492688475, −0.873506291583413989050938032970, 2.24469506527201965320212075000, 3.17741844820625456058121066502, 3.61041743696542012345012642623, 5.12421792787509530665346932822, 6.05609412425595251721090393120, 6.44317810373634674756527309279, 7.56729301203215177465814703552, 8.824604849029398637699163204044, 9.806101045590175710723562573789, 10.67473397910154811907560092032

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