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Properties

Label 2-819-117.49-c1-0-49
Degree $2$
Conductor $819$
Sign $-0.745 - 0.666i$
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  + (−1.46 − 0.844i)2-s + (−1.54 + 0.781i)3-s + (0.425 + 0.737i)4-s + (−1.72 − 0.997i)5-s + (2.92 + 0.162i)6-s − i·7-s + 1.93i·8-s + (1.77 − 2.41i)9-s + (1.68 + 2.91i)10-s + (−2.41 − 1.39i)11-s + (−1.23 − 0.807i)12-s + (2.53 − 2.56i)13-s + (−0.844 + 1.46i)14-s + (3.45 + 0.192i)15-s + (2.48 − 4.31i)16-s + (1.71 − 2.97i)17-s + ⋯
L(s)  = 1  + (−1.03 − 0.597i)2-s + (−0.892 + 0.450i)3-s + (0.212 + 0.368i)4-s + (−0.773 − 0.446i)5-s + (1.19 + 0.0665i)6-s − 0.377i·7-s + 0.685i·8-s + (0.593 − 0.805i)9-s + (0.532 + 0.923i)10-s + (−0.727 − 0.420i)11-s + (−0.356 − 0.233i)12-s + (0.702 − 0.711i)13-s + (−0.225 + 0.390i)14-s + (0.891 + 0.0497i)15-s + (0.622 − 1.07i)16-s + (0.416 − 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0542483 + 0.142172i\)
\(L(\frac12)\) \(\approx\) \(0.0542483 + 0.142172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 0.781i)T \)
7 \( 1 + iT \)
13 \( 1 + (-2.53 + 2.56i)T \)
good2 \( 1 + (1.46 + 0.844i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.72 + 0.997i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.41 + 1.39i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.71 + 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 - 1.47i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.868T + 23T^{2} \)
29 \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.38 + 3.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.93 + 3.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.25iT - 41T^{2} \)
43 \( 1 - 2.62T + 43T^{2} \)
47 \( 1 + (4.45 - 2.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.32T + 53T^{2} \)
59 \( 1 + (10.8 - 6.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.66T + 61T^{2} \)
67 \( 1 - 2.09iT - 67T^{2} \)
71 \( 1 + (8.10 + 4.68i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + (-1.52 - 2.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 + 5.83i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (15.1 - 8.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.73iT - 97T^{2} \)
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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

−9.820496832100147751765943574442, −9.126495190698070145628471807531, −8.003566252479593108276977849223, −7.60894936109207466692260360830, −6.01731440227609612721348521853, −5.26921896615547285821161467993, −4.22875570075144142925463220899, −3.04845113441202438062918180863, −1.11025854141495286605957800566, −0.15038126234471561606832332563, 1.55385182795781385573996866786, 3.40784203735211469414811886209, 4.63995466883469135410310949050, 5.86624084812824385460431453264, 6.66082075110377284495906462705, 7.48506606770499179452050956636, 7.934935334044294825487930908385, 8.927919842143956460502552368709, 9.873701127334427749306922241192, 10.74828803827490589634171907117

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