Properties

Label 2-819-117.50-c1-0-4
Degree $2$
Conductor $819$
Sign $0.614 - 0.789i$
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  + (−0.548 − 2.04i)2-s + (−1.71 + 0.237i)3-s + (−2.15 + 1.24i)4-s + (0.530 + 1.97i)5-s + (1.42 + 3.38i)6-s + (0.707 − 0.707i)7-s + (0.733 + 0.733i)8-s + (2.88 − 0.814i)9-s + (3.75 − 2.17i)10-s + (0.0834 − 0.0223i)11-s + (3.40 − 2.64i)12-s + (−0.572 − 3.55i)13-s + (−1.83 − 1.05i)14-s + (−1.37 − 3.26i)15-s + (−1.39 + 2.40i)16-s + (−1.73 + 3.00i)17-s + ⋯
L(s)  = 1  + (−0.387 − 1.44i)2-s + (−0.990 + 0.137i)3-s + (−1.07 + 0.622i)4-s + (0.237 + 0.885i)5-s + (0.582 + 1.38i)6-s + (0.267 − 0.267i)7-s + (0.259 + 0.259i)8-s + (0.962 − 0.271i)9-s + (1.18 − 0.686i)10-s + (0.0251 − 0.00674i)11-s + (0.982 − 0.764i)12-s + (−0.158 − 0.987i)13-s + (−0.490 − 0.283i)14-s + (−0.356 − 0.844i)15-s + (−0.347 + 0.602i)16-s + (−0.420 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.614 - 0.789i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286783 + 0.140220i\)
\(L(\frac12)\) \(\approx\) \(0.286783 + 0.140220i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.237i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.572 + 3.55i)T \)
good2 \( 1 + (0.548 + 2.04i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.530 - 1.97i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0834 + 0.0223i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.73 - 3.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.99 - 2.14i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 + (-7.84 - 4.52i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.614 - 0.164i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.94 + 1.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.725 + 0.725i)T - 41iT^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 + (2.14 - 8.00i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 6.66iT - 53T^{2} \)
59 \( 1 + (2.80 - 10.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + (4.35 + 4.35i)T + 67iT^{2} \)
71 \( 1 + (-4.03 - 15.0i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.701 - 0.701i)T - 73iT^{2} \)
79 \( 1 + (1.21 + 2.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.700 - 0.187i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.28 - 8.53i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.72 - 1.72i)T + 97iT^{2} \)
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

−10.45638489063638500795923179297, −10.24327699249804899382328367552, −8.966852561949506080017508836827, −7.990764580444120169076198454893, −6.65776201116634182114593101288, −6.12056823501036844818063100966, −4.70504116835540897361693635497, −3.78662978746605874433636523988, −2.63015456075542156930882011254, −1.42595032731803229735873988835, 0.20721262116908167371854385459, 1.99942047994146484252739799974, 4.56499184030155111108961515945, 4.84570608923968656043895819068, 5.97718163990634096146824167247, 6.57327485569245162877100649461, 7.31598917416171394189715465859, 8.470239840621473652881222241950, 8.916655264947645489549012215609, 9.857504546167125378253457544649

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