↑

Properties

Label 2-819-91.4-c1-0-29
Degree $2$
Conductor $819$
Sign $0.894 + 0.446i$
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  + 0.180i·2-s + 1.96·4-s + (2.32 − 1.34i)5-s + (−2.46 − 0.967i)7-s + 0.717i·8-s + (0.242 + 0.420i)10-s + (−2.33 + 1.34i)11-s + (1.92 − 3.05i)13-s + (0.174 − 0.445i)14-s + 3.80·16-s + 4.76·17-s + (0.163 + 0.0942i)19-s + (4.57 − 2.64i)20-s + (−0.243 − 0.421i)22-s + 4.39·23-s + ⋯
L(s)  = 1  + 0.127i·2-s + 0.983·4-s + (1.04 − 0.600i)5-s + (−0.930 − 0.365i)7-s + 0.253i·8-s + (0.0768 + 0.133i)10-s + (−0.703 + 0.406i)11-s + (0.532 − 0.846i)13-s + (0.0467 − 0.119i)14-s + 0.951·16-s + 1.15·17-s + (0.0374 + 0.0216i)19-s + (1.02 − 0.590i)20-s + (−0.0519 − 0.0899i)22-s + 0.917·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10038 - 0.495396i\)
\(L(\frac12)\) \(\approx\) \(2.10038 - 0.495396i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.967i)T \)
13 \( 1 + (-1.92 + 3.05i)T \)
good2 \( 1 - 0.180iT - 2T^{2} \)
5 \( 1 + (-2.32 + 1.34i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.33 - 1.34i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + (-0.163 - 0.0942i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.20 + 1.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.95iT - 37T^{2} \)
41 \( 1 + (4.70 + 2.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.00 + 6.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.60 + 0.924i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.53 - 6.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.58iT - 59T^{2} \)
61 \( 1 + (-0.205 + 0.356i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.87 - 5.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.89 - 1.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.3 - 7.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 - 5.89iT - 89T^{2} \)
97 \( 1 + (-0.390 + 0.225i)T + (48.5 - 84.0i)T^{2} \)
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

−10.14212437186720471013664974738, −9.595896725663425113685596532664, −8.385445607452686683130092658421, −7.53785862910488279421464419426, −6.61586165034948532881038009125, −5.79596814741993553205141125226, −5.16770277635535457066811516264, −3.47309750725431807294526361320, −2.54112554313928427341224084084, −1.18173748128029979898072282486, 1.60519038849866369859789861956, 2.79904311032790012841988304807, 3.37284160948896068161952134982, 5.28774435618838807935328594475, 6.09511841334352621769903018028, 6.65796423901832074788225570621, 7.50374256776652502462009446435, 8.768233260409144391636382252792, 9.637923589592102803767469136142, 10.38117093697116760229865080317

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