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Properties

Label 2-819-91.67-c0-0-0
Degree $2$
Conductor $819$
Sign $0.818 + 0.575i$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)4-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + (−0.366 + 0.366i)19-s + (−0.866 − 0.5i)25-s − 0.999i·28-s + (0.366 + 1.36i)31-s + (−0.5 + 0.133i)37-s + (1.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s + 1.73·61-s − 0.999i·64-s + (−1 + i)67-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + (−0.366 + 0.366i)19-s + (−0.866 − 0.5i)25-s − 0.999i·28-s + (0.366 + 1.36i)31-s + (−0.5 + 0.133i)37-s + (1.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s + 1.73·61-s − 0.999i·64-s + (−1 + i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.818 + 0.575i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.818 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185739203\)
\(L(\frac12)\) \(\approx\) \(1.185739203\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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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

−10.30880754041816921736122443143, −9.851412206101690271138308278140, −8.596045092680636548029194366197, −7.57247009896421151528779520251, −7.00413122506102861104146279792, −6.08236340971301555016360185405, −5.02020567025159185866521503885, −4.04986296281894392663774182877, −2.62153557837044334587783572346, −1.45823378368166458461027255327, 2.02337167280871345113185974527, 2.80366753766539992535523215536, 4.11377982116627186934721578177, 5.36327450569760800441968400123, 6.12184918493857249235578019908, 7.25704756397129989719478031650, 7.86298234241593969977573841782, 8.729992692397280035212316987105, 9.677000167191598151991163238460, 10.66829304722149074251141703441

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