↑

Properties

Label 2-819-91.89-c1-0-35
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: $\chi_{819} (271, \cdot )$
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

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

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