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Properties

Label 2-819-117.2-c1-0-23
Degree $2$
Conductor $819$
Sign $-0.856 - 0.516i$
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  + (1.78 + 1.78i)2-s + (−1.55 − 0.759i)3-s + 4.39i·4-s + (0.310 − 1.15i)5-s + (−1.42 − 4.14i)6-s + (0.258 − 0.965i)7-s + (−4.28 + 4.28i)8-s + (1.84 + 2.36i)9-s + (2.62 − 1.51i)10-s + (−4.23 + 4.23i)11-s + (3.33 − 6.84i)12-s + (3.51 + 0.790i)13-s + (2.19 − 1.26i)14-s + (−1.36 + 1.56i)15-s − 6.52·16-s + (−2.05 + 3.56i)17-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)2-s + (−0.898 − 0.438i)3-s + 2.19i·4-s + (0.138 − 0.517i)5-s + (−0.582 − 1.69i)6-s + (0.0978 − 0.365i)7-s + (−1.51 + 1.51i)8-s + (0.615 + 0.787i)9-s + (0.830 − 0.479i)10-s + (−1.27 + 1.27i)11-s + (0.963 − 1.97i)12-s + (0.975 + 0.219i)13-s + (0.585 − 0.337i)14-s + (−0.351 + 0.404i)15-s − 1.63·16-s + (−0.499 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.856 - 0.516i$
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: \((2,\ 819,\ (\ :1/2),\ -0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530027 + 1.90327i\)
\(L(\frac12)\) \(\approx\) \(0.530027 + 1.90327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.55 + 0.759i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-3.51 - 0.790i)T \)
good2 \( 1 + (-1.78 - 1.78i)T + 2iT^{2} \)
5 \( 1 + (-0.310 + 1.15i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.23 - 4.23i)T - 11iT^{2} \)
17 \( 1 + (2.05 - 3.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 4.39i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.09 - 7.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.35iT - 29T^{2} \)
31 \( 1 + (2.14 + 0.575i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.19 + 4.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.16 + 1.38i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.241 + 0.139i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.837 + 3.12i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 + (-1.72 + 1.72i)T - 59iT^{2} \)
61 \( 1 + (-2.24 - 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 - 12.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.79 + 0.749i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.56 + 6.56i)T + 73iT^{2} \)
79 \( 1 + (-4.95 + 8.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.32 + 1.96i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (12.6 + 3.37i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (13.6 + 3.67i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.75936729492327100784041454431, −9.803851711383405260917879722644, −8.302568374715527114087269847911, −7.65118518561575284420585779505, −7.02919643232833173335289172082, −5.91233712203317300262411008485, −5.54293235467836372865620074164, −4.55866310574806741489075569243, −3.83792558772851837211151076397, −1.87863490576864456064414627349, 0.72671975955352792711312909514, 2.58195060735232950902616059892, 3.24856711648070096738610480034, 4.48336565360402499606840899940, 5.22485105652217482704663945018, 5.95946612329781917732993504645, 6.71201792978263500680915957545, 8.405679472636970677223933201748, 9.500335712039508454239495857071, 10.54914657510582009603109606270

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