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Properties

Label 2-819-91.74-c1-0-25
Degree $2$
Conductor $819$
Sign $0.927 + 0.374i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + (1.5 + 2.59i)5-s + (2 − 1.73i)7-s + 3·8-s + (−1.5 − 2.59i)10-s + (−1.5 − 2.59i)11-s + (−1 − 3.46i)13-s + (−2 + 1.73i)14-s − 16-s + 2·17-s + (0.5 − 0.866i)19-s + (−1.5 − 2.59i)20-s + (1.5 + 2.59i)22-s + (−2 + 3.46i)25-s + (1 + 3.46i)26-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s + (0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s + 1.06·8-s + (−0.474 − 0.821i)10-s + (−0.452 − 0.783i)11-s + (−0.277 − 0.960i)13-s + (−0.534 + 0.462i)14-s − 0.250·16-s + 0.485·17-s + (0.114 − 0.198i)19-s + (−0.335 − 0.580i)20-s + (0.319 + 0.553i)22-s + (−0.400 + 0.692i)25-s + (0.196 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06401 - 0.206852i\)
\(L(\frac12)\) \(\approx\) \(1.06401 - 0.206852i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.5 - 11.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19498515193657594743587468463, −9.561407804816850601755642414400, −8.300126757796985391135939740893, −7.85170734237166521008911248876, −6.94443125250656373439691471542, −5.79822073224444378540588248686, −4.90940484260439897130647317915, −3.62128743612012108477951849056, −2.43760343044770223921198018033, −0.835655756923601098163415042913, 1.23821931987795033679240901262, 2.18038063981077940198230753869, 4.23956269856436241918243244399, 4.99660545715066328729164023005, 5.58625417441522171535448808730, 7.10504011516982068271075081864, 8.044422464239794806373274107640, 8.746883799490829594903641028401, 9.343161768955426914645143453045, 9.939165597226645645218230901820

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