↑

Properties

Label 2-819-91.5-c1-0-26
Degree $2$
Conductor $819$
Sign $-0.226 + 0.974i$
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  + (−0.423 − 1.58i)2-s + (−0.587 + 0.339i)4-s + (2.77 − 0.742i)5-s + (1.92 + 1.81i)7-s + (−1.52 − 1.52i)8-s + (−2.34 − 4.06i)10-s + (−0.894 + 3.33i)11-s + (2.29 − 2.78i)13-s + (2.04 − 3.81i)14-s + (−2.44 + 4.24i)16-s + (−3.22 − 5.58i)17-s + (2.93 − 0.786i)19-s + (−1.37 + 1.37i)20-s + 5.65·22-s + (2.97 + 1.71i)23-s + ⋯
L(s)  = 1  + (−0.299 − 1.11i)2-s + (−0.293 + 0.169i)4-s + (1.24 − 0.332i)5-s + (0.729 + 0.684i)7-s + (−0.540 − 0.540i)8-s + (−0.742 − 1.28i)10-s + (−0.269 + 1.00i)11-s + (0.635 − 0.771i)13-s + (0.546 − 1.02i)14-s + (−0.612 + 1.06i)16-s + (−0.782 − 1.35i)17-s + (0.673 − 0.180i)19-s + (−0.307 + 0.307i)20-s + 1.20·22-s + (0.619 + 0.357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14558 - 1.44210i\)
\(L(\frac12)\) \(\approx\) \(1.14558 - 1.44210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.92 - 1.81i)T \)
13 \( 1 + (-2.29 + 2.78i)T \)
good2 \( 1 + (0.423 + 1.58i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-2.77 + 0.742i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.894 - 3.33i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.93 + 0.786i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.97 - 1.71i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.40T + 29T^{2} \)
31 \( 1 + (-0.868 + 3.24i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.75 + 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.03 - 3.03i)T + 41iT^{2} \)
43 \( 1 + 4.48iT - 43T^{2} \)
47 \( 1 + (-1.47 - 5.51i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.72 + 9.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.86 + 0.766i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.03 + 1.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.04 + 1.88i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.31 - 8.31i)T - 71iT^{2} \)
73 \( 1 + (-4.51 - 1.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.543 - 0.942i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.01 + 2.01i)T + 83iT^{2} \)
89 \( 1 + (1.28 + 4.79i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.20 + 3.20i)T + 97iT^{2} \)
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

−9.883768932069134225716793043636, −9.446006828729789184617416465913, −8.738958464086424814393912543962, −7.53295973229556110084386161523, −6.36306650634355110014110847655, −5.43696036547294707900277459852, −4.63150385217979659073763226613, −2.90981588871635927263630641159, −2.20732438927043559962325008466, −1.16009222594572915076236365462, 1.52972221701100770669920315339, 2.89445623921124883296197166306, 4.40198687567011507478622941820, 5.60082818593139237802073820981, 6.23791230453066013472978206803, 6.86459281371466941975375628965, 7.945802346746611307041711837721, 8.615541576922540415000505136571, 9.347309450382666343253747971745, 10.60113239301301105443721862128

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