↑

Properties

Label 2-819-117.2-c1-0-19
Degree $2$
Conductor $819$
Sign $0.392 + 0.919i$
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  + (−1.45 − 1.45i)2-s + (−0.883 − 1.48i)3-s + 2.20i·4-s + (0.260 − 0.972i)5-s + (−0.879 + 3.44i)6-s + (−0.258 + 0.965i)7-s + (0.301 − 0.301i)8-s + (−1.43 + 2.63i)9-s + (−1.78 + 1.03i)10-s + (−1.42 + 1.42i)11-s + (3.28 − 1.95i)12-s + (3.42 − 1.12i)13-s + (1.77 − 1.02i)14-s + (−1.67 + 0.470i)15-s + 3.54·16-s + (−3.04 + 5.27i)17-s + ⋯
L(s)  = 1  + (−1.02 − 1.02i)2-s + (−0.510 − 0.860i)3-s + 1.10i·4-s + (0.116 − 0.434i)5-s + (−0.359 + 1.40i)6-s + (−0.0978 + 0.365i)7-s + (0.106 − 0.106i)8-s + (−0.479 + 0.877i)9-s + (−0.565 + 0.326i)10-s + (−0.430 + 0.430i)11-s + (0.949 − 0.563i)12-s + (0.949 − 0.313i)13-s + (0.474 − 0.274i)14-s + (−0.433 + 0.121i)15-s + 0.885·16-s + (−0.738 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.392 + 0.919i$
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: \((2,\ 819,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523748 - 0.345921i\)
\(L(\frac12)\) \(\approx\) \(0.523748 - 0.345921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.883 + 1.48i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-3.42 + 1.12i)T \)
good2 \( 1 + (1.45 + 1.45i)T + 2iT^{2} \)
5 \( 1 + (-0.260 + 0.972i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.42 - 1.42i)T - 11iT^{2} \)
17 \( 1 + (3.04 - 5.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.568 + 2.12i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.726 + 1.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.669iT - 29T^{2} \)
31 \( 1 + (-7.78 - 2.08i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.67 - 6.24i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.98 + 2.13i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.20 - 1.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.873 + 3.25i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 2.67iT - 53T^{2} \)
59 \( 1 + (5.30 - 5.30i)T - 59iT^{2} \)
61 \( 1 + (-4.43 - 7.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.64 + 6.14i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.37 + 2.51i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 73iT^{2} \)
79 \( 1 + (-0.958 + 1.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.21 + 0.862i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.61 - 1.50i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.5 - 3.08i)T + (84.0 + 48.5i)T^{2} \)
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

−10.43817964104624780949043974220, −9.125537877488792400060789459427, −8.537322577123645549825313643924, −7.904638393028408073269022028639, −6.63426670926890104462868149848, −5.82865143573303062586293572269, −4.70013791706886557670445743530, −3.02599535638847634072335083066, −1.97427657531129296908910923044, −0.975887376135064127127864241205, 0.62172497472248017594034368397, 2.99497240320190548528822801224, 4.17966119377847344301289459080, 5.37407524332329684889614848475, 6.30397505253021078180980115240, 6.81557880479179714039899649944, 7.918763490614131870760521463801, 8.762313870613791137944895137069, 9.456512333780798205354823154357, 10.19282108284372859912471231536

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