↑

Properties

Label 2-819-91.45-c1-0-18
Degree $2$
Conductor $819$
Sign $0.994 - 0.104i$
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  + (1.48 − 1.48i)2-s − 2.39i·4-s + (−0.507 + 1.89i)5-s + (−0.313 + 2.62i)7-s + (−0.588 − 0.588i)8-s + (2.05 + 3.55i)10-s + (−0.648 + 2.42i)11-s + (3.33 + 1.36i)13-s + (3.43 + 4.35i)14-s + 3.04·16-s − 7.32·17-s + (0.930 − 0.249i)19-s + (4.53 + 1.21i)20-s + (2.62 + 4.55i)22-s + 6.63i·23-s + ⋯
L(s)  = 1  + (1.04 − 1.04i)2-s − 1.19i·4-s + (−0.226 + 0.846i)5-s + (−0.118 + 0.992i)7-s + (−0.207 − 0.207i)8-s + (0.649 + 1.12i)10-s + (−0.195 + 0.730i)11-s + (0.925 + 0.378i)13-s + (0.916 + 1.16i)14-s + 0.762·16-s − 1.77·17-s + (0.213 − 0.0572i)19-s + (1.01 + 0.271i)20-s + (0.560 + 0.970i)22-s + 1.38i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.994 - 0.104i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.994 - 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48563 + 0.130772i\)
\(L(\frac12)\) \(\approx\) \(2.48563 + 0.130772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.313 - 2.62i)T \)
13 \( 1 + (-3.33 - 1.36i)T \)
good2 \( 1 + (-1.48 + 1.48i)T - 2iT^{2} \)
5 \( 1 + (0.507 - 1.89i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.648 - 2.42i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 7.32T + 17T^{2} \)
19 \( 1 + (-0.930 + 0.249i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.63iT - 23T^{2} \)
29 \( 1 + (-5.21 + 9.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.92 - 0.782i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.974 + 0.974i)T + 37iT^{2} \)
41 \( 1 + (0.710 - 0.190i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.1 + 5.84i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.62 - 0.971i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.15 + 2.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.35 + 3.35i)T - 59iT^{2} \)
61 \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 1.62i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.32 + 2.23i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.01 - 7.50i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.31 + 4.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + (-2.89 + 2.89i)T - 89iT^{2} \)
97 \( 1 + (2.63 - 9.85i)T + (-84.0 - 48.5i)T^{2} \)
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

−10.68072791768638090895158490120, −9.611975704612488349958326189711, −8.767873743730873897508616878524, −7.57387185377381581558348772413, −6.52018783479802534750357135490, −5.67425704157564357799579795369, −4.59361229336758567746205180460, −3.73254012816940826067986899374, −2.68886827846092601930523500870, −1.94353749420818467928187968452, 0.926590983442268335635914805669, 3.15659211177510765291600584591, 4.29732255054753091810622131453, 4.69840171105343962035059741955, 5.87809972981672291302773060018, 6.60084581937480655893657466071, 7.40691361399306532952869285831, 8.448245886618062067499655909857, 8.907219225316161969838087973729, 10.53765868838689049988206864991

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