Properties

Label 2-819-117.2-c1-0-44
Degree $2$
Conductor $819$
Sign $-0.698 + 0.715i$
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  + (−1.80 − 1.80i)2-s + (−0.649 + 1.60i)3-s + 4.54i·4-s + (0.921 − 3.43i)5-s + (4.07 − 1.72i)6-s + (−0.258 + 0.965i)7-s + (4.60 − 4.60i)8-s + (−2.15 − 2.08i)9-s + (−7.88 + 4.55i)10-s + (4.12 − 4.12i)11-s + (−7.29 − 2.95i)12-s + (2.00 − 2.99i)13-s + (2.21 − 1.27i)14-s + (4.92 + 3.71i)15-s − 7.57·16-s + (−2.65 + 4.59i)17-s + ⋯
L(s)  = 1  + (−1.27 − 1.27i)2-s + (−0.374 + 0.927i)3-s + 2.27i·4-s + (0.412 − 1.53i)5-s + (1.66 − 0.706i)6-s + (−0.0978 + 0.365i)7-s + (1.62 − 1.62i)8-s + (−0.718 − 0.695i)9-s + (−2.49 + 1.44i)10-s + (1.24 − 1.24i)11-s + (−2.10 − 0.852i)12-s + (0.556 − 0.831i)13-s + (0.592 − 0.341i)14-s + (1.27 + 0.958i)15-s − 1.89·16-s + (−0.642 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262382 - 0.622859i\)
\(L(\frac12)\) \(\approx\) \(0.262382 - 0.622859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.649 - 1.60i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-2.00 + 2.99i)T \)
good2 \( 1 + (1.80 + 1.80i)T + 2iT^{2} \)
5 \( 1 + (-0.921 + 3.43i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.12 + 4.12i)T - 11iT^{2} \)
17 \( 1 + (2.65 - 4.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.48 - 5.55i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.288 + 0.499i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.42iT - 29T^{2} \)
31 \( 1 + (-1.47 - 0.394i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.40 + 5.23i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.89 - 1.31i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.10 + 2.94i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.91 + 10.8i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 1.68iT - 53T^{2} \)
59 \( 1 + (-9.92 + 9.92i)T - 59iT^{2} \)
61 \( 1 + (-0.0677 - 0.117i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.64 + 6.12i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.69 - 1.79i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.00 + 5.00i)T + 73iT^{2} \)
79 \( 1 + (-2.37 + 4.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.62 - 1.77i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.88 - 1.04i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.22 - 0.862i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.910956551790210930266115729115, −9.102456259794050533942039844697, −8.614459616876166169784370091176, −8.246716730707523110732652784483, −6.20705938616274402593141635606, −5.43338801504270092859817846953, −4.01008825477814315780559973084, −3.42518994701762086861537199012, −1.67530723191803298960537987968, −0.61936012299502468209656522765, 1.30287274395245690694607401398, 2.55509895888232404937063558550, 4.62083820570686541808531050511, 6.02506805665695254862751096198, 6.72851055781098641799422503757, 6.97157514665429702013608538550, 7.53186440268489231446609273096, 8.866266435742296419070660190994, 9.510809895636822133968202399363, 10.28980115140285194622063102749

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