Properties

Label 2-819-91.2-c0-0-0
Degree $2$
Conductor $819$
Sign $0.467 - 0.884i$
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  + i·4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)13-s − 16-s + (0.5 + 0.133i)19-s + (0.866 − 0.5i)25-s + (−0.5 + 0.866i)28-s + (−1.36 − 0.366i)31-s + (0.366 + 0.366i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s i·64-s + (−0.366 − 1.36i)67-s + ⋯
L(s)  = 1  + i·4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)13-s − 16-s + (0.5 + 0.133i)19-s + (0.866 − 0.5i)25-s + (−0.5 + 0.866i)28-s + (−1.36 − 0.366i)31-s + (0.366 + 0.366i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (−0.866 − 1.5i)61-s i·64-s + (−0.366 − 1.36i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.467 - 0.884i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (730, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.467 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018087575\)
\(L(\frac12)\) \(\approx\) \(1.018087575\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
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 \( 1 + iT^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
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

−10.81221757945136576433777930049, −9.474683452929971466347792091887, −8.920243341890646112675628599459, −7.895346977098269944492767465347, −7.44263098902453268022039894309, −6.32081880523776641423373293890, −5.06868123916336621326220287199, −4.33702109191745518472809897415, −3.07170183736791462023445336777, −2.01939365502429053450448712524, 1.18537092265828073204698414618, 2.51422642931971795856576461271, 4.09155526514782354611348265603, 5.11254612560912582179466153225, 5.64262534157675612940134875579, 7.00775148587544687291411407312, 7.53716428980118435737982812071, 8.748658300376336932787269998767, 9.507256784530401311334466114653, 10.45115370678247060667368600669

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