↑

Properties

Label 2-684-19.12-c4-0-0
Degree $2$
Conductor $684$
Sign $0.780 - 0.624i$
Analytic cond. $70.7050$
Root an. cond. $8.40862$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.7 − 20.3i)5-s − 65.4·7-s − 235.·11-s + (−173. − 100. i)13-s + (−251. − 436. i)17-s + (−97.9 + 347. i)19-s + (253. − 439. i)23-s + (35.8 − 62.1i)25-s + (−273. − 158. i)29-s + 1.05e3i·31-s + (769. + 1.33e3i)35-s + 700. i·37-s + (−826. + 477. i)41-s + (209. + 363. i)43-s + (1.20e3 − 2.08e3i)47-s + ⋯
L(s)  = 1  + (−0.470 − 0.814i)5-s − 1.33·7-s − 1.94·11-s + (−1.02 − 0.593i)13-s + (−0.871 − 1.50i)17-s + (−0.271 + 0.962i)19-s + (0.479 − 0.830i)23-s + (0.0574 − 0.0994i)25-s + (−0.325 − 0.187i)29-s + 1.10i·31-s + (0.628 + 1.08i)35-s + 0.512i·37-s + (−0.491 + 0.283i)41-s + (0.113 + 0.196i)43-s + (0.545 − 0.944i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.780 - 0.624i$
Analytic conductor: \(70.7050\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :2),\ 0.780 - 0.624i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.06204523467\)
\(L(\frac12)\) \(\approx\) \(0.06204523467\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (97.9 - 347. i)T \)
good5 \( 1 + (11.7 + 20.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 65.4T + 2.40e3T^{2} \)
11 \( 1 + 235.T + 1.46e4T^{2} \)
13 \( 1 + (173. + 100. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (251. + 436. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-253. + 439. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (273. + 158. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.05e3iT - 9.23e5T^{2} \)
37 \( 1 - 700. iT - 1.87e6T^{2} \)
41 \( 1 + (826. - 477. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-209. - 363. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.20e3 + 2.08e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (3.71e3 + 2.14e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-1.41e3 + 816. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (939. - 1.62e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (391. + 225. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-1.06e3 + 617. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.19e3 + 5.53e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-9.44e3 + 5.45e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 635.T + 4.74e7T^{2} \)
89 \( 1 + (6.25e3 + 3.61e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-4.79e3 + 2.76e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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

−10.05930200315523978281761575187, −9.142713255856478099513065599683, −8.225393421290250949306665263472, −7.44919294980640584454384965442, −6.50855384683680251542398371017, −5.21905768466186943204192830083, −4.73333163712856606668843357291, −3.20819469189144989555207344474, −2.46624777114123059469910490492, −0.39237199516314851786418928504, 0.03557103431978824667543022420, 2.30964309820932804405650210635, 3.01298437591831690693137774060, 4.11495167779396024326149828670, 5.32554652117253745223933741579, 6.39690342644627775181178827958, 7.16866563362848695528176220668, 7.85241669222753572365926728960, 9.078095861675327871509309297408, 9.860167690602079862172095539173

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