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Properties

Label 2-819-91.81-c1-0-5
Degree $2$
Conductor $819$
Sign $-0.851 - 0.523i$
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  + (−0.857 + 1.48i)2-s + (−0.470 − 0.815i)4-s + (−1.22 − 2.12i)5-s + (2.18 + 1.49i)7-s − 1.81·8-s + 4.21·10-s + 1.03·11-s + (−3.36 + 1.29i)13-s + (−4.09 + 1.95i)14-s + (2.49 − 4.32i)16-s + (1.50 + 2.61i)17-s + 3.19·19-s + (−1.15 + 2.00i)20-s + (−0.890 + 1.54i)22-s + (−1.73 + 3.00i)23-s + ⋯
L(s)  = 1  + (−0.606 + 1.05i)2-s + (−0.235 − 0.407i)4-s + (−0.549 − 0.951i)5-s + (0.824 + 0.565i)7-s − 0.641·8-s + 1.33·10-s + 0.313·11-s + (−0.932 + 0.360i)13-s + (−1.09 + 0.523i)14-s + (0.624 − 1.08i)16-s + (0.366 + 0.634i)17-s + 0.732·19-s + (−0.258 + 0.447i)20-s + (−0.189 + 0.328i)22-s + (−0.361 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.851 - 0.523i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.851 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229390 + 0.811193i\)
\(L(\frac12)\) \(\approx\) \(0.229390 + 0.811193i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 1.49i)T \)
13 \( 1 + (3.36 - 1.29i)T \)
good2 \( 1 + (0.857 - 1.48i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
17 \( 1 + (-1.50 - 2.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (1.73 - 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.01 - 6.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 - 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.54 - 4.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.21 - 5.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.88 + 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.90 - 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.47 - 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.60T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (-1.63 + 2.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.50 + 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.211 + 0.366i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.34T + 83T^{2} \)
89 \( 1 + (-2.65 + 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.92 + 5.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.35005315041666968914737727400, −9.230869369583087052410028478010, −8.792762266343608686909030873353, −7.981033027786649876566786222253, −7.44351496355214634596007240406, −6.38220828928001095125317503607, −5.29814672687667802717146865163, −4.68902849009911478705472430455, −3.20235396923180568121581631381, −1.42942637659312802826033175704, 0.53846120171215439535674616875, 2.09622343197798029410947677713, 3.07543800649510045710879705013, 4.08105570223560153843152255014, 5.33551425642783420646636107050, 6.62070655911806316738537464849, 7.53996597278309918578046885432, 8.151282297432681774819375019144, 9.410915252724036679133274815176, 10.03650736303827118604241603510

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