↑

Properties

Label 2-819-7.2-c1-0-39
Degree $2$
Conductor $819$
Sign $-0.608 - 0.793i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 2.36i)2-s + (−2.71 − 4.69i)4-s + (1.09 − 1.89i)5-s + (−2.19 + 1.47i)7-s − 9.33·8-s + (−2.98 − 5.16i)10-s + (−0.524 − 0.907i)11-s + 13-s + (0.484 + 7.19i)14-s + (−7.29 + 12.6i)16-s + (−2.64 − 4.58i)17-s + (−0.378 + 0.655i)19-s − 11.8·20-s − 2.85·22-s + (0.326 − 0.566i)23-s + ⋯
L(s)  = 1  + (0.963 − 1.66i)2-s + (−1.35 − 2.34i)4-s + (0.489 − 0.847i)5-s + (−0.830 + 0.557i)7-s − 3.30·8-s + (−0.942 − 1.63i)10-s + (−0.158 − 0.273i)11-s + 0.277·13-s + (0.129 + 1.92i)14-s + (−1.82 + 3.16i)16-s + (−0.641 − 1.11i)17-s + (−0.0868 + 0.150i)19-s − 2.65·20-s − 0.609·22-s + (0.0681 − 0.118i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796320 + 1.61411i\)
\(L(\frac12)\) \(\approx\) \(0.796320 + 1.61411i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 1.47i)T \)
13 \( 1 - T \)
good2 \( 1 + (-1.36 + 2.36i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.524 + 0.907i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.64 + 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.378 - 0.655i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.326 + 0.566i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.10T + 29T^{2} \)
31 \( 1 + (0.513 + 0.890i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.44 + 9.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 - 0.887T + 43T^{2} \)
47 \( 1 + (-1.16 + 2.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.524 + 0.907i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.24 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.23 + 3.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.60T + 71T^{2} \)
73 \( 1 + (-4.14 - 7.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.07 - 1.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.66T + 83T^{2} \)
89 \( 1 + (2.88 - 4.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.88T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.696467483608795906668149052832, −9.323468891936888298493607912214, −8.544044587708975612607949612642, −6.63894781148425697860443756386, −5.64700126406870887425270298295, −5.07988887735436483834803221518, −4.03448045348364326218715405269, −2.96933909354590169025950693890, −2.07806099317137135277096499044, −0.63687257994219050547929030194, 2.79272861244120994395449572551, 3.74985029461224987741588712220, 4.64339041382421102423861127760, 5.82078031640372348578465421500, 6.57080815653259832699897288808, 6.85940094460222465675404456790, 7.938761422520236553183409769852, 8.740023986139112994046630652374, 9.822447322121454847047727481354, 10.63489592376342458641368860436

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