↑

Properties

Label 2-819-91.16-c1-0-23
Degree $2$
Conductor $819$
Sign $0.401 + 0.916i$
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  − 2.55·2-s + 4.50·4-s + (1.39 − 2.41i)5-s + (−2.46 − 0.968i)7-s − 6.39·8-s + (−3.55 + 6.15i)10-s + (−1.38 + 2.39i)11-s + (2.99 + 2.01i)13-s + (6.28 + 2.46i)14-s + 7.28·16-s + 5.88·17-s + (−1.70 − 2.94i)19-s + (6.27 − 10.8i)20-s + (3.52 − 6.11i)22-s + 7.34·23-s + ⋯
L(s)  = 1  − 1.80·2-s + 2.25·4-s + (0.623 − 1.07i)5-s + (−0.930 − 0.365i)7-s − 2.25·8-s + (−1.12 + 1.94i)10-s + (−0.417 + 0.722i)11-s + (0.830 + 0.557i)13-s + (1.67 + 0.659i)14-s + 1.82·16-s + 1.42·17-s + (−0.390 − 0.675i)19-s + (1.40 − 2.43i)20-s + (0.752 − 1.30i)22-s + 1.53·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.401 + 0.916i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.401 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561713 - 0.367250i\)
\(L(\frac12)\) \(\approx\) \(0.561713 - 0.367250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.968i)T \)
13 \( 1 + (-2.99 - 2.01i)T \)
good2 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 + (-1.39 + 2.41i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.38 - 2.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 + (1.70 + 2.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.34T + 23T^{2} \)
29 \( 1 + (1.56 + 2.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 - 3.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 + (3.24 + 5.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 + 5.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.95 - 6.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.34 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.73T + 59T^{2} \)
61 \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 7.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.57 + 2.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.80 + 8.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.88 - 3.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.82T + 83T^{2} \)
89 \( 1 + 1.75T + 89T^{2} \)
97 \( 1 + (-8.48 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.758991843276110003276828556553, −9.312421149594459314223868552872, −8.667801899538227400142046620519, −7.71155480561139933096636256720, −6.88541413347777353373055929110, −6.05383097829936017760803138700, −4.84622153604359856750109892077, −3.17699638571402430324718340857, −1.78034405490905666421876820743, −0.72485473202671129580102995660, 1.13514198303750923600903580682, 2.74412336188989428172654756858, 3.21663091253788526355191304686, 5.78512547725808455985628866975, 6.21146634130955336348450682550, 7.13931049880646931660687220685, 7.993991280474946150781231703684, 8.781790002029147908248677756495, 9.681199942992456757691497794671, 10.20150802389144540562567076633

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