Properties

Label 2-819-91.51-c1-0-31
Degree $2$
Conductor $819$
Sign $0.273 - 0.961i$
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  + (1.97 + 1.14i)2-s + (1.61 + 2.78i)4-s + (1.84 + 1.06i)5-s + (2.62 − 0.331i)7-s + 2.78i·8-s + (2.42 + 4.20i)10-s + (0.267 − 0.154i)11-s + (−3.22 − 1.62i)13-s + (5.57 + 2.34i)14-s + (0.0349 − 0.0605i)16-s + (0.887 + 1.53i)17-s + (−1.54 − 0.890i)19-s + 6.84i·20-s + 0.704·22-s + (−0.575 + 0.996i)23-s + ⋯
L(s)  = 1  + (1.39 + 0.807i)2-s + (0.805 + 1.39i)4-s + (0.823 + 0.475i)5-s + (0.992 − 0.125i)7-s + 0.985i·8-s + (0.767 + 1.32i)10-s + (0.0805 − 0.0465i)11-s + (−0.893 − 0.449i)13-s + (1.48 + 0.626i)14-s + (0.00874 − 0.0151i)16-s + (0.215 + 0.372i)17-s + (−0.353 − 0.204i)19-s + 1.53i·20-s + 0.150·22-s + (−0.119 + 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.273 - 0.961i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.14823 + 2.37697i\)
\(L(\frac12)\) \(\approx\) \(3.14823 + 2.37697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.331i)T \)
13 \( 1 + (3.22 + 1.62i)T \)
good2 \( 1 + (-1.97 - 1.14i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.84 - 1.06i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.267 + 0.154i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.887 - 1.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.54 + 0.890i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.575 - 0.996i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.01T + 29T^{2} \)
31 \( 1 + (3.98 - 2.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.79 + 2.77i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.72iT - 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (8.24 + 4.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.72 - 6.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.03 - 4.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 + 2.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.9 + 6.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 + (10.2 - 5.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.96 + 6.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (-1.43 - 0.829i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.66iT - 97T^{2} \)
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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

−10.49114778348066052422921578185, −9.670003244329081620655245804421, −8.377006060225793884983649490436, −7.51036784201785939413592646198, −6.78159057579936187696087080501, −5.83126824532443733056495598792, −5.19879818837073922335884590862, −4.34191879333593729431887614501, −3.17431564602402249239197887931, −1.97450397282439625263412296383, 1.64718164359679784002275051720, 2.31444183539141909737493970613, 3.70077984836778078834861614493, 4.78919750626774647281837479297, 5.22397319304862148972854198044, 6.11104901689965079337222203326, 7.31829248395076953568799453665, 8.478040574714841089647856150891, 9.461801705393905650091924105435, 10.28065813171798719376997804592

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