Properties

Label 2-819-117.103-c1-0-26
Degree $2$
Conductor $819$
Sign $-0.116 - 0.993i$
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  + (0.223 − 0.129i)2-s + (−1.68 + 0.404i)3-s + (−0.966 + 1.67i)4-s + (1.67 + 0.965i)5-s + (−0.324 + 0.308i)6-s + (−0.866 + 0.5i)7-s + 1.01i·8-s + (2.67 − 1.36i)9-s + 0.499·10-s + (2.98 − 1.72i)11-s + (0.951 − 3.21i)12-s + (3.60 + 0.0177i)13-s + (−0.129 + 0.223i)14-s + (−3.20 − 0.950i)15-s + (−1.80 − 3.12i)16-s + 6.09·17-s + ⋯
L(s)  = 1  + (0.158 − 0.0913i)2-s + (−0.972 + 0.233i)3-s + (−0.483 + 0.837i)4-s + (0.748 + 0.431i)5-s + (−0.132 + 0.125i)6-s + (−0.327 + 0.188i)7-s + 0.359i·8-s + (0.891 − 0.453i)9-s + 0.157·10-s + (0.898 − 0.518i)11-s + (0.274 − 0.926i)12-s + (0.999 + 0.00492i)13-s + (−0.0345 + 0.0598i)14-s + (−0.828 − 0.245i)15-s + (−0.450 − 0.780i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.116 - 0.993i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826179 + 0.928902i\)
\(L(\frac12)\) \(\approx\) \(0.826179 + 0.928902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.404i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.60 - 0.0177i)T \)
good2 \( 1 + (-0.223 + 0.129i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.67 - 0.965i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.98 + 1.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.09T + 17T^{2} \)
19 \( 1 - 4.74iT - 19T^{2} \)
23 \( 1 + (4.12 - 7.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.17 - 5.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.08 + 2.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.74iT - 37T^{2} \)
41 \( 1 + (10.5 + 6.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.993 - 1.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.36 + 1.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.86T + 53T^{2} \)
59 \( 1 + (3.12 + 1.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 - 8.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.24iT - 71T^{2} \)
73 \( 1 + 16.8iT - 73T^{2} \)
79 \( 1 + (-2.30 - 3.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.48 - 4.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + (2.60 - 1.50i)T + (48.5 - 84.0i)T^{2} \)
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

−10.34270579461959652311115220423, −9.797134971409539259051134210691, −8.895874804601387059581722151479, −7.901376069874784714253829369255, −6.81173595149065310149665465966, −5.90233898945442729409125957703, −5.39917405616469252990121714874, −3.83706235167196412517252627314, −3.43058360177617526084870251174, −1.47717304699423988446628405352, 0.76202773792527383110206723810, 1.77635229699173860965936506244, 3.90254018696112953848089503341, 4.82307272802931172736086368691, 5.69426277137708089604420258960, 6.28010986322278395434701360245, 7.03998510465776706419387338706, 8.458531583094056946595881681599, 9.446832801668087293457191252531, 9.999021240414140806747846549171

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