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Properties

Label 2-819-91.19-c1-0-36
Degree $2$
Conductor $819$
Sign $0.797 - 0.603i$
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  + (2.64 + 0.707i)2-s + (4.74 + 2.73i)4-s + (0.792 − 0.212i)5-s + (−1.19 − 2.36i)7-s + (6.71 + 6.71i)8-s + 2.24·10-s + (−0.566 − 0.566i)11-s + (2.33 − 2.75i)13-s + (−1.48 − 7.07i)14-s + (7.51 + 13.0i)16-s + (−0.145 + 0.251i)17-s + (1.50 + 1.50i)19-s + (4.33 + 1.16i)20-s + (−1.09 − 1.89i)22-s + (−8.07 + 4.66i)23-s + ⋯
L(s)  = 1  + (1.86 + 0.500i)2-s + (2.37 + 1.36i)4-s + (0.354 − 0.0949i)5-s + (−0.451 − 0.892i)7-s + (2.37 + 2.37i)8-s + 0.708·10-s + (−0.170 − 0.170i)11-s + (0.646 − 0.763i)13-s + (−0.397 − 1.89i)14-s + (1.87 + 3.25i)16-s + (−0.0352 + 0.0610i)17-s + (0.345 + 0.345i)19-s + (0.969 + 0.259i)20-s + (−0.233 − 0.404i)22-s + (−1.68 + 0.972i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.797 - 0.603i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.797 - 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.55836 + 1.53015i\)
\(L(\frac12)\) \(\approx\) \(4.55836 + 1.53015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 2.36i)T \)
13 \( 1 + (-2.33 + 2.75i)T \)
good2 \( 1 + (-2.64 - 0.707i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.792 + 0.212i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.566 + 0.566i)T + 11iT^{2} \)
17 \( 1 + (0.145 - 0.251i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.50 - 1.50i)T + 19iT^{2} \)
23 \( 1 + (8.07 - 4.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.87 - 4.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.35 + 5.03i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.591 + 2.20i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.80 - 2.35i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.71 + 1.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.25 + 4.70i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.69 - 4.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.22 + 12.0i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.84iT - 61T^{2} \)
67 \( 1 + (-8.97 + 8.97i)T - 67iT^{2} \)
71 \( 1 + (-6.43 - 1.72i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.74 + 2.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.75 + 4.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.51 - 3.51i)T + 83iT^{2} \)
89 \( 1 + (-4.25 - 1.14i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.20 - 11.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.61196452447709091126591875782, −9.693677112572310227472073221309, −8.052108546118422481172336746630, −7.55793620654269827559022341284, −6.48633105341003081115529316458, −5.83834210094192244786104038995, −5.08769195414815297250995017783, −3.77855290422413552724940755233, −3.45978384964732826588782680942, −1.91091766114138572245936142029, 1.88144690096572030137828302320, 2.66101417192752081995068824823, 3.78972037998077144166237073226, 4.64278839007097749264015044717, 5.75593612125224850462749319977, 6.18481346922826877048233671204, 7.03819475268996002657499868295, 8.433280420749709366123691686430, 9.733935997351105332263981529931, 10.31051404152062941922574652547

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