Properties

Label 2-684-19.12-c4-0-28
Degree $2$
Conductor $684$
Sign $-0.629 + 0.776i$
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  + (23.1 + 40.0i)5-s − 21.4·7-s + 119.·11-s + (−150. − 86.9i)13-s + (−190. − 329. i)17-s + (21.6 − 360. i)19-s + (−448. + 776. i)23-s + (−758. + 1.31e3i)25-s + (−1.34e3 − 775. i)29-s + 796. i·31-s + (−495. − 858. i)35-s + 1.52e3i·37-s + (1.22e3 − 707. i)41-s + (−1.22e3 − 2.11e3i)43-s + (2.05e3 − 3.56e3i)47-s + ⋯
L(s)  = 1  + (0.925 + 1.60i)5-s − 0.437·7-s + 0.990·11-s + (−0.891 − 0.514i)13-s + (−0.658 − 1.14i)17-s + (0.0599 − 0.998i)19-s + (−0.847 + 1.46i)23-s + (−1.21 + 2.10i)25-s + (−1.59 − 0.922i)29-s + 0.829i·31-s + (−0.404 − 0.701i)35-s + 1.11i·37-s + (0.729 − 0.421i)41-s + (−0.660 − 1.14i)43-s + (0.931 − 1.61i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.629 + 0.776i$
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: \((2,\ 684,\ (\ :2),\ -0.629 + 0.776i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1578435852\)
\(L(\frac12)\) \(\approx\) \(0.1578435852\)
\(L(3)\) 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 \( 1 + (-21.6 + 360. i)T \)
good5 \( 1 + (-23.1 - 40.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 21.4T + 2.40e3T^{2} \)
11 \( 1 - 119.T + 1.46e4T^{2} \)
13 \( 1 + (150. + 86.9i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (190. + 329. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (448. - 776. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.34e3 + 775. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 796. iT - 9.23e5T^{2} \)
37 \( 1 - 1.52e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.22e3 + 707. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.22e3 + 2.11e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.05e3 + 3.56e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (399. + 230. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.06e3 - 1.76e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-766. + 1.32e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (963. + 556. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (639. - 369. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-1.86e3 - 3.22e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.16e3 - 674. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 9.02e3T + 4.74e7T^{2} \)
89 \( 1 + (3.69e3 + 2.13e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.36e4 + 7.87e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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

−9.699063682833415481323858779971, −9.068677267519243564795062798989, −7.39278993906979719647953561233, −7.01003186575553153087164807493, −6.13017723068614563759726221107, −5.23045405725608772648528977849, −3.70310003751797352932419821215, −2.80042647383084232877641481591, −1.91160682513754298195389229538, −0.03395374560827363750846746184, 1.38043313316083664567670536064, 2.15086989004103499077972939233, 3.99555665844762068861746058595, 4.63255451508941882706174500400, 5.88602906434516994806142254354, 6.32194244320432751309940887900, 7.74701453728731668681705801537, 8.715494774841991359524428757881, 9.354978716416092496484404070086, 9.863410722410678149613547786029

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