Properties

Label 2-819-91.59-c1-0-15
Degree $2$
Conductor $819$
Sign $0.709 + 0.704i$
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  + (−1.92 − 1.92i)2-s + 5.43i·4-s + (3.41 + 0.914i)5-s + (2.45 + 0.996i)7-s + (6.62 − 6.62i)8-s + (−4.82 − 8.34i)10-s + (0.426 + 0.114i)11-s + (3.60 + 0.0874i)13-s + (−2.80 − 6.64i)14-s − 14.6·16-s − 1.43·17-s + (0.340 + 1.27i)19-s + (−4.97 + 18.5i)20-s + (−0.602 − 1.04i)22-s − 7.18i·23-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)2-s + 2.71i·4-s + (1.52 + 0.409i)5-s + (0.926 + 0.376i)7-s + (2.34 − 2.34i)8-s + (−1.52 − 2.64i)10-s + (0.128 + 0.0344i)11-s + (0.999 + 0.0242i)13-s + (−0.749 − 1.77i)14-s − 3.67·16-s − 0.347·17-s + (0.0782 + 0.291i)19-s + (−1.11 + 4.15i)20-s + (−0.128 − 0.222i)22-s − 1.49i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09156 - 0.449598i\)
\(L(\frac12)\) \(\approx\) \(1.09156 - 0.449598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 0.996i)T \)
13 \( 1 + (-3.60 - 0.0874i)T \)
good2 \( 1 + (1.92 + 1.92i)T + 2iT^{2} \)
5 \( 1 + (-3.41 - 0.914i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.426 - 0.114i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 1.43T + 17T^{2} \)
19 \( 1 + (-0.340 - 1.27i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 7.18iT - 23T^{2} \)
29 \( 1 + (3.82 - 6.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.187 - 0.698i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.719 + 0.719i)T - 37iT^{2} \)
41 \( 1 + (-0.748 - 2.79i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.20 + 4.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.242 - 0.906i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.48 - 4.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.29 - 2.29i)T + 59iT^{2} \)
61 \( 1 + (11.2 + 6.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.64 + 6.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.487 - 1.82i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (12.2 - 3.29i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.77 + 3.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.33 - 2.33i)T - 83iT^{2} \)
89 \( 1 + (7.39 + 7.39i)T + 89iT^{2} \)
97 \( 1 + (2.51 + 0.673i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.33223136597729490180862017301, −9.215181542955147114441443990068, −8.900591043953065376854740261473, −8.007358742139626274904162536468, −6.90394662971722016649731935505, −5.84422320972295251414942873699, −4.41801103438579116645292318425, −3.03966974005146400548547796525, −2.09959147181711478617559408350, −1.35266420786500499567526906149, 1.13938431408449656876705208629, 1.94513025416145520559946620218, 4.51320022798343922063215511392, 5.65754720105857729433931052394, 5.91336799648507489759019259633, 7.02748599089040659717361584596, 7.85590376774934322714696348117, 8.683733917438413668781638203366, 9.333471282285239728241055140350, 9.919971963384863720682799309099

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