↑

Properties

Label 2-1638-7.2-c1-0-19
Degree $2$
Conductor $1638$
Sign $0.605 - 0.795i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.22 − 2.12i)5-s + (2.5 − 0.866i)7-s + 0.999·8-s + (1.22 + 2.12i)10-s + (2.72 + 4.71i)11-s + 13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (−3.17 + 5.49i)19-s − 2.44·20-s − 5.44·22-s + (−3.67 + 6.36i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.547 − 0.948i)5-s + (0.944 − 0.327i)7-s + 0.353·8-s + (0.387 + 0.670i)10-s + (0.821 + 1.42i)11-s + 0.277·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.728 + 1.26i)19-s − 0.547·20-s − 1.16·22-s + (−0.766 + 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.787857787\)
\(L(\frac12)\) \(\approx\) \(1.787857787\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.72 - 4.71i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.17 - 5.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 - 6.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.550T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.67 + 9.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.44T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (3.94 - 6.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.27 - 5.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.17 - 8.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.17 - 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 + (1.67 + 2.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-6.67 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.34T + 97T^{2} \)
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

−9.315662849040550480371449981685, −8.800999422664865430280266560234, −7.67314252349454166552762686658, −7.51043650144728733165988270959, −6.03039421135937279989509506924, −5.67760915387751648700950784494, −4.43690149576541605198227895765, −4.08694874466484453533647653451, −1.85467624510192722844120702399, −1.36207009956885090022652931503, 0.883517240008078940239894759411, 2.25834058727028760913886654549, 2.93953520563579134165063424765, 4.06287480908571370132747366702, 5.08391976442599958273783749230, 6.22026548319176795244693913419, 6.70978293219354600616577695703, 7.951997600913260585089116220371, 8.583547202446185140313248563831, 9.223227711313394405246809382195

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