Properties

Label 2-819-117.2-c1-0-14
Degree $2$
Conductor $819$
Sign $0.295 - 0.955i$
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  + (−1.39 − 1.39i)2-s + (−1.02 + 1.39i)3-s + 1.88i·4-s + (−0.512 + 1.91i)5-s + (3.37 − 0.519i)6-s + (−0.258 + 0.965i)7-s + (−0.163 + 0.163i)8-s + (−0.901 − 2.86i)9-s + (3.37 − 1.95i)10-s + (2.85 − 2.85i)11-s + (−2.62 − 1.92i)12-s + (1.33 + 3.34i)13-s + (1.70 − 0.985i)14-s + (−2.14 − 2.67i)15-s + 4.22·16-s + (3.82 − 6.62i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.985i)2-s + (−0.591 + 0.806i)3-s + 0.941i·4-s + (−0.229 + 0.855i)5-s + (1.37 − 0.211i)6-s + (−0.0978 + 0.365i)7-s + (−0.0577 + 0.0577i)8-s + (−0.300 − 0.953i)9-s + (1.06 − 0.616i)10-s + (0.859 − 0.859i)11-s + (−0.759 − 0.556i)12-s + (0.371 + 0.928i)13-s + (0.456 − 0.263i)14-s + (−0.554 − 0.690i)15-s + 1.05·16-s + (0.927 − 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443826 + 0.327299i\)
\(L(\frac12)\) \(\approx\) \(0.443826 + 0.327299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.02 - 1.39i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-1.33 - 3.34i)T \)
good2 \( 1 + (1.39 + 1.39i)T + 2iT^{2} \)
5 \( 1 + (0.512 - 1.91i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.85 + 2.85i)T - 11iT^{2} \)
17 \( 1 + (-3.82 + 6.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.147 + 0.551i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.82 - 6.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0304iT - 29T^{2} \)
31 \( 1 + (-6.35 - 1.70i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.15 - 8.05i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (9.95 - 2.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.11 - 4.68i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.357 + 1.33i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 1.72iT - 53T^{2} \)
59 \( 1 + (2.55 - 2.55i)T - 59iT^{2} \)
61 \( 1 + (1.19 + 2.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.73 - 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.52 + 0.676i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.71 + 2.71i)T + 73iT^{2} \)
79 \( 1 + (1.33 - 2.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.38 - 0.370i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.13 - 1.10i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.4 - 3.88i)T + (84.0 + 48.5i)T^{2} \)
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

−10.26303161497332682713335472689, −9.768265790368011331087319105810, −9.081104824105911901212183277751, −8.275135749937878018065691607430, −6.91900439970770868489565389568, −6.10091602902233943212098890083, −5.00815214366747582450981349635, −3.51147156476977026003021165336, −3.02629325973729102237104060915, −1.25084979840873975503303424223, 0.48423326261066871743475586538, 1.61462250207021547784990108389, 3.77066394356888000094773037005, 5.03756659788858988757703365773, 6.12421419678518362755861227255, 6.61236641761707842002753539938, 7.65693776178696894255235552827, 8.237221086370835206275876702075, 8.778883381767428934644357544113, 10.12740856082444218685163047136

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