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Properties

Label 2-819-91.45-c1-0-34
Degree $2$
Conductor $819$
Sign $0.540 + 0.841i$
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  + (−1.12 + 1.12i)2-s − 0.509i·4-s + (0.973 − 3.63i)5-s + (2.28 − 1.33i)7-s + (−1.66 − 1.66i)8-s + (2.97 + 5.15i)10-s + (−0.872 + 3.25i)11-s + (−3.03 − 1.94i)13-s + (−1.06 + 4.05i)14-s + 4.75·16-s + 3.33·17-s + (0.733 − 0.196i)19-s + (−1.84 − 0.495i)20-s + (−2.67 − 4.62i)22-s − 3.56i·23-s + ⋯
L(s)  = 1  + (−0.791 + 0.791i)2-s − 0.254i·4-s + (0.435 − 1.62i)5-s + (0.863 − 0.504i)7-s + (−0.590 − 0.590i)8-s + (0.942 + 1.63i)10-s + (−0.263 + 0.982i)11-s + (−0.842 − 0.538i)13-s + (−0.284 + 1.08i)14-s + 1.18·16-s + 0.809·17-s + (0.168 − 0.0451i)19-s + (−0.413 − 0.110i)20-s + (−0.569 − 0.986i)22-s − 0.742i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802973 - 0.438286i\)
\(L(\frac12)\) \(\approx\) \(0.802973 - 0.438286i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 + 1.33i)T \)
13 \( 1 + (3.03 + 1.94i)T \)
good2 \( 1 + (1.12 - 1.12i)T - 2iT^{2} \)
5 \( 1 + (-0.973 + 3.63i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.872 - 3.25i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.33T + 17T^{2} \)
19 \( 1 + (-0.733 + 0.196i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.56iT - 23T^{2} \)
29 \( 1 + (-1.42 + 2.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.90 - 2.38i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (8.01 + 8.01i)T + 37iT^{2} \)
41 \( 1 + (6.84 - 1.83i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.9 + 6.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.32 + 0.356i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.59 + 6.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.23 + 2.23i)T - 59iT^{2} \)
61 \( 1 + (-0.902 - 0.521i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.85 - 0.765i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.00 - 1.87i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.559 - 2.08i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.54 + 9.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.51 - 1.51i)T + 83iT^{2} \)
89 \( 1 + (-2.23 + 2.23i)T - 89iT^{2} \)
97 \( 1 + (3.23 - 12.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.833277096017053810769827829528, −9.070591109152693348515143722639, −8.375252320620737228716857033929, −7.64714242619855374515745885483, −7.04071983343693524081471714123, −5.51055956895926838127229786774, −5.05281800231782061666766423643, −3.91923274688369532616170868894, −1.96572739550423382629166766806, −0.59023735485902670255379036296, 1.63616537754054252854654849076, 2.60263846189579082405265620291, 3.40161668300669542346865354170, 5.27907830521814575644905487007, 5.92155378791041653492007282822, 7.09727268855008158093547737151, 7.931910680880133544436320894156, 8.918033292648064523444527283885, 9.724548280512552768437354842466, 10.42251986982553406142250396893

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