↑

Properties

Label 2-2646-63.59-c1-0-18
Degree $2$
Conductor $2646$
Sign $0.0945 - 0.995i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 1.79·5-s + 0.999i·8-s + (1.55 + 0.895i)10-s + 2.40i·11-s + (4.23 + 2.44i)13-s + (−0.5 + 0.866i)16-s + (−1.83 + 3.17i)17-s + (2.61 − 1.50i)19-s + (0.895 + 1.55i)20-s + (−1.20 + 2.07i)22-s + 3.76i·23-s − 1.79·25-s + (2.44 + 4.23i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.800·5-s + 0.353i·8-s + (0.490 + 0.283i)10-s + 0.724i·11-s + (1.17 + 0.678i)13-s + (−0.125 + 0.216i)16-s + (−0.444 + 0.769i)17-s + (0.599 − 0.346i)19-s + (0.200 + 0.346i)20-s + (−0.256 + 0.443i)22-s + 0.785i·23-s − 0.358·25-s + (0.479 + 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0945 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0945 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.0945 - 0.995i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.0945 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.110498738\)
\(L(\frac12)\) \(\approx\) \(3.110498738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.79T + 5T^{2} \)
11 \( 1 - 2.40iT - 11T^{2} \)
13 \( 1 + (-4.23 - 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.61 + 1.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.76iT - 23T^{2} \)
29 \( 1 + (-5.68 + 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.02 - 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.68 + 8.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.04 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.48 + 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.29 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.81 - 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.285 + 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + (-10.7 - 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.51 - 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.00 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.87 - 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 - 2.75i)T + (48.5 - 84.0i)T^{2} \)
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

−8.906053635025881483688145102011, −8.355572564907021513301505431650, −7.21275819817092223176343161079, −6.73355082798211019183183037367, −5.84717421719454554228414405419, −5.33125394388660817069901827859, −4.24580458335722827244056174258, −3.61109967668983428303206005427, −2.33288461521862502645069064946, −1.51283902297897526473468344099, 0.843885358102788008044102545697, 1.96201277812643311818477078761, 3.04194555689262234174144297702, 3.66926303188464129986354190145, 4.89292140798099437987233859188, 5.49854981753948090145002622915, 6.26622219356038700665887051743, 6.83339150096789316205454305647, 8.075309196344764063760702022513, 8.673956664974106220952672418988

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