↑

Properties

Label 2-819-91.16-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.776 + 0.630i$
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  − 0.0680·2-s − 1.99·4-s + (−1.52 + 2.64i)5-s + (0.910 + 2.48i)7-s + 0.271·8-s + (0.104 − 0.180i)10-s + (−2.17 + 3.77i)11-s + (−1.79 − 3.12i)13-s + (−0.0619 − 0.169i)14-s + 3.97·16-s + 3.52·17-s + (−3.45 − 5.97i)19-s + (3.05 − 5.28i)20-s + (0.148 − 0.256i)22-s − 3.33·23-s + ⋯
L(s)  = 1  − 0.0481·2-s − 0.997·4-s + (−0.684 + 1.18i)5-s + (0.344 + 0.938i)7-s + 0.0961·8-s + (0.0329 − 0.0569i)10-s + (−0.656 + 1.13i)11-s + (−0.499 − 0.866i)13-s + (−0.0165 − 0.0451i)14-s + 0.993·16-s + 0.855·17-s + (−0.791 − 1.37i)19-s + (0.682 − 1.18i)20-s + (0.0316 − 0.0547i)22-s − 0.695·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.776 + 0.630i$
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: \((2,\ 819,\ (\ :1/2),\ -0.776 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0528566 - 0.149006i\)
\(L(\frac12)\) \(\approx\) \(0.0528566 - 0.149006i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.910 - 2.48i)T \)
13 \( 1 + (1.79 + 3.12i)T \)
good2 \( 1 + 0.0680T + 2T^{2} \)
5 \( 1 + (1.52 - 2.64i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.17 - 3.77i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + (3.45 + 5.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.33T + 23T^{2} \)
29 \( 1 + (4.95 + 8.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.62 - 8.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.109T + 37T^{2} \)
41 \( 1 + (1.76 + 3.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.844 - 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.28 - 2.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.65 + 4.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.55T + 59T^{2} \)
61 \( 1 + (2.43 + 4.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.340 - 0.589i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.61 + 4.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 - 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.85 + 8.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.41T + 83T^{2} \)
89 \( 1 - 7.70T + 89T^{2} \)
97 \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.55174671449346276017220389809, −9.996877118252637855935276496570, −9.097014642698668712917660849237, −7.971306279338583178613624815706, −7.66115222537524412451127193642, −6.44398898796230908076529019468, −5.27031633531138273038482922189, −4.57256214646716062135586438055, −3.30909090271237954074208113093, −2.34955615195671736135004871519, 0.087260281665605644328103614436, 1.35406371550538683406558965348, 3.58166548661370905152894936551, 4.25043687113665899216739181921, 5.04248232285474164231986305718, 5.97087607956435010807108546213, 7.57318076168118928841792268606, 8.113831004006884546912006177296, 8.691495593289869329657614259924, 9.679958364881141630547642086724

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