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Properties

Label 2-819-91.87-c0-0-0
Degree $2$
Conductor $819$
Sign $0.617 - 0.786i$
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  − 4-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + 16-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s + 37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)52-s + (1.5 + 0.866i)61-s − 64-s + (−1 − 1.73i)67-s + (−1.5 − 0.866i)73-s + ⋯
L(s)  = 1  − 4-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + 16-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s + 37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)52-s + (1.5 + 0.866i)61-s − 64-s + (−1 − 1.73i)67-s + (−1.5 − 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8020642283\)
\(L(\frac12)\) \(\approx\) \(0.8020642283\)
\(L(1)\) 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 \( 1 + T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
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 \( 1 - T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
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

−10.37715619235657258397931742278, −9.488258459446268291836736491896, −9.056641003085510335428444256714, −8.076046721512629579379304849504, −7.34816052787656851050030506288, −5.92036844969497696192874352788, −5.24181018929504839678816978479, −4.34875285848319683333620110328, −3.20971600105167329798203868666, −1.68220092236419185657992676657, 0.955619624188024603504560698777, 2.91263491552876245492436884355, 4.08179559491327690540716560168, 4.88767605507628185783348369126, 5.68892897177952016853665238410, 7.08733919732790301419221204933, 7.84325022560625444612728023527, 8.531755787475903630705867945162, 9.720116675462312843151978574982, 10.02803754812366738123815326105

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