Properties

Label 2-819-91.3-c0-0-0
Degree $2$
Conductor $819$
Sign $0.923 - 0.384i$
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  + (0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s − 1.73i·19-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s − 1.73i·61-s − 0.999·64-s + 2·67-s + (−1.5 − 0.866i)73-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s − 1.73i·19-s + (−0.5 + 0.866i)25-s + 0.999·28-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s − 1.73i·61-s − 0.999·64-s + 2·67-s + (−1.5 − 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (640, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.923 - 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141167590\)
\(L(\frac12)\) \(\approx\) \(1.141167590\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.77273491722723459770137964034, −9.571791563743384835220266351617, −8.698949477041907137226865956737, −7.87579694828925260051585480360, −7.04899505417660130099545624888, −6.48854271412159783648826316568, −4.95538752550857059906230171658, −4.08310327443642008501866969417, −3.08883804952070340749989553130, −1.71960731527707592749539533693, 1.53086773007200169560309015155, 2.62269463602989264999895534160, 4.03204835574637248451053451884, 5.54769636772491087569489348714, 5.66816164337314419103386480039, 6.81686260175457226496083490539, 7.965146770870713980278781464670, 8.603685693725296909688919642642, 9.745529986161441691621849835214, 10.36651498786995526246140185940

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