Properties

Label 2-819-91.11-c0-0-0
Degree $2$
Conductor $819$
Sign $0.999 + 0.0247i$
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 + (1.36 + 1.36i)19-s + (0.866 + 0.5i)25-s + 0.999i·28-s + (−1.36 + 0.366i)31-s + (−0.5 − 1.86i)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 + (1.36 + 1.36i)19-s + (0.866 + 0.5i)25-s + 0.999i·28-s + (−1.36 + 0.366i)31-s + (−0.5 − 1.86i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0247i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.999 + 0.0247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8971553877\)
\(L(\frac12)\) \(\approx\) \(0.8971553877\)
\(L(1)\) 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 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−10.52403060717540282138395632216, −9.463322661929802573631553809019, −8.789720302073274423239527270075, −7.71872498564144079265791293213, −7.43447111586758402498875781759, −5.88050261215556366916036047167, −5.04829324274203416652598710843, −3.93846544203856168203041988128, −3.31052947040954895929487667957, −1.27520525454857803587190793091, 1.39005318553599220502305882319, 2.94263895845517460930360134371, 4.28500326620099445323270361678, 5.14099150499034411808459670238, 5.85227941283572077116123002544, 6.95155218175250706796369691965, 8.119360106353005796258367289655, 9.027783125758584661260911466670, 9.248762769368219574260648237105, 10.45655616663492799850900290603

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