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Properties

Label 2-819-91.54-c1-0-25
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: $\chi_{819} (145, \cdot )$
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

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

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