Properties

Label 2-819-273.263-c0-0-1
Degree $2$
Conductor $819$
Sign $0.804 - 0.594i$
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  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−1.22 + 0.707i)17-s − 19-s + (−0.5 + 0.866i)25-s + 1.41i·26-s + 0.999·28-s + (−1.22 + 0.707i)29-s + (1.22 − 0.707i)32-s − 2·34-s + (0.5 − 0.866i)37-s + (−1.22 − 0.707i)38-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−1.22 + 0.707i)17-s − 19-s + (−0.5 + 0.866i)25-s + 1.41i·26-s + 0.999·28-s + (−1.22 + 0.707i)29-s + (1.22 − 0.707i)32-s − 2·34-s + (0.5 − 0.866i)37-s + (−1.22 − 0.707i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.804 - 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.807096503\)
\(L(\frac12)\) \(\approx\) \(1.807096503\)
\(L(1)\) 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 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−10.89876744032102958920478075189, −9.632698104539057229610274857923, −8.681530606514609707971179405337, −7.63302932267221848232131016177, −6.84737266431481646257831683377, −6.18618248190775560570579753769, −5.13110239423132635615231063600, −4.18310729161215574148627627533, −3.73159561459754316324706148078, −1.88266440907859191654634450334, 2.01517580466560765505959420727, 2.80491011594458792785395835629, 4.03859387462722706650639064039, 4.83191163344377785297471016451, 5.72201359632170442579964116992, 6.46699843213494934159912514244, 7.987359392886177191369313169476, 8.578808829597014890917097954924, 9.667887844073859160576643101546, 10.79746292349459032333622270316

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