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Properties

Label 2-819-91.10-c0-0-0
Degree $2$
Conductor $819$
Sign $0.113 + 0.993i$
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 − 19-s + (0.5 − 0.866i)25-s − 0.999·28-s + (1 − 1.73i)31-s + (1.5 + 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 + (−0.5 + 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 − 19-s + (0.5 − 0.866i)25-s − 0.999·28-s + (1 − 1.73i)31-s + (1.5 + 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 + (−0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.113 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8493996027\)
\(L(\frac12)\) \(\approx\) \(0.8493996027\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.17814672943098997207796086068, −9.675379484901362270071254992516, −8.444979694669830980357865937862, −7.85680207937478145750331635329, −6.66204851508183731140613624775, −5.84654372093183091949817798488, −4.72627144520745610558122706915, −4.18208811522284662874452033387, −2.51336804413697191267273628089, −0.934737568045017834870138961393, 2.06649556262421308201651612768, 3.19964359488687969167508652742, 4.43539684240640698274596304489, 5.08439988137484896117598942996, 6.37142346767501235042248267585, 7.31359582596072007924328658084, 8.248597631803731126062408275754, 8.886070916929263907906506235250, 9.511271702219506773307400488829, 10.73291220258166499968369529921

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