↑

Properties

Label 2-684-19.12-c4-0-9
Degree $2$
Conductor $684$
Sign $-0.603 - 0.797i$
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  + (23.2 + 40.3i)5-s + 0.442·7-s − 11.3·11-s + (−108. − 62.4i)13-s + (90.7 + 157. i)17-s + (344. + 107. i)19-s + (314. − 544. i)23-s + (−770. + 1.33e3i)25-s + (1.20e3 + 698. i)29-s + 1.42e3i·31-s + (10.3 + 17.8i)35-s − 566. i·37-s + (−1.67e3 + 968. i)41-s + (619. + 1.07e3i)43-s + (144. − 250. i)47-s + ⋯
L(s)  = 1  + (0.930 + 1.61i)5-s + 0.00903·7-s − 0.0941·11-s + (−0.640 − 0.369i)13-s + (0.313 + 0.543i)17-s + (0.955 + 0.296i)19-s + (0.594 − 1.02i)23-s + (−1.23 + 2.13i)25-s + (1.43 + 0.830i)29-s + 1.48i·31-s + (0.00841 + 0.0145i)35-s − 0.413i·37-s + (−0.997 + 0.575i)41-s + (0.335 + 0.580i)43-s + (0.0655 − 0.113i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.603 - 0.797i$
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: \((2,\ 684,\ (\ :2),\ -0.603 - 0.797i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.190073049\)
\(L(\frac12)\) \(\approx\) \(2.190073049\)
\(L(3)\) 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 \( 1 + (-344. - 107. i)T \)
good5 \( 1 + (-23.2 - 40.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 0.442T + 2.40e3T^{2} \)
11 \( 1 + 11.3T + 1.46e4T^{2} \)
13 \( 1 + (108. + 62.4i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-90.7 - 157. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-314. + 544. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.20e3 - 698. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.42e3iT - 9.23e5T^{2} \)
37 \( 1 + 566. iT - 1.87e6T^{2} \)
41 \( 1 + (1.67e3 - 968. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-619. - 1.07e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-144. + 250. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-3.46e3 - 1.99e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-81.2 + 46.8i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (560. - 970. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.27e3 + 1.31e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-1.32e3 + 764. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (4.26e3 + 7.38e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.75e3 - 3.32e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.98e3T + 4.74e7T^{2} \)
89 \( 1 + (1.20e4 + 6.97e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-3.70e3 + 2.13e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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

−10.40825347119257017316837748942, −9.592099250233828603843807436556, −8.469712891782106510711642307527, −7.32515700591548525525560383902, −6.72164150657373061612546164944, −5.86025517380119188506577923972, −4.88368632839399425228733097443, −3.24341862999636715476942967305, −2.72655837067896716693866212720, −1.41080072926073332253458733173, 0.51946862882478870239683567473, 1.49292416703564499780520143581, 2.66712398929767954175922629547, 4.26282646388241086852893842593, 5.14440143702393460976133238383, 5.68562074315621732204107715851, 6.94931212535922773886500286804, 8.000474222489362075723865597449, 8.856984648120242941278925359067, 9.657600280372529865355785190802

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