↑

Properties

Label 2-624-12.11-c3-0-18
Degree $2$
Conductor $624$
Sign $-0.997 + 0.0687i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.90 + 4.31i)3-s + 9.14i·5-s + 18.5i·7-s + (−10.1 + 25.0i)9-s − 8.04·11-s + 13·13-s + (−39.4 + 26.5i)15-s + 37.3i·17-s + 42.4i·19-s + (−79.8 + 53.7i)21-s − 19.9·23-s + 41.4·25-s + (−137. + 28.7i)27-s + 9.94i·29-s + 30.2i·31-s + ⋯
L(s)  = 1  + (0.558 + 0.829i)3-s + 0.817i·5-s + 0.999i·7-s + (−0.376 + 0.926i)9-s − 0.220·11-s + 0.277·13-s + (−0.678 + 0.456i)15-s + 0.533i·17-s + 0.512i·19-s + (−0.829 + 0.558i)21-s − 0.180·23-s + 0.331·25-s + (−0.978 + 0.204i)27-s + 0.0637i·29-s + 0.175i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0687i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.997 + 0.0687i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.997 + 0.0687i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.811530787\)
\(L(\frac12)\) \(\approx\) \(1.811530787\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.90 - 4.31i)T \)
13 \( 1 - 13T \)
good5 \( 1 - 9.14iT - 125T^{2} \)
7 \( 1 - 18.5iT - 343T^{2} \)
11 \( 1 + 8.04T + 1.33e3T^{2} \)
17 \( 1 - 37.3iT - 4.91e3T^{2} \)
19 \( 1 - 42.4iT - 6.85e3T^{2} \)
23 \( 1 + 19.9T + 1.21e4T^{2} \)
29 \( 1 - 9.94iT - 2.43e4T^{2} \)
31 \( 1 - 30.2iT - 2.97e4T^{2} \)
37 \( 1 + 116.T + 5.06e4T^{2} \)
41 \( 1 + 308. iT - 6.89e4T^{2} \)
43 \( 1 - 26.8iT - 7.95e4T^{2} \)
47 \( 1 + 111.T + 1.03e5T^{2} \)
53 \( 1 + 331. iT - 1.48e5T^{2} \)
59 \( 1 - 327.T + 2.05e5T^{2} \)
61 \( 1 + 130.T + 2.26e5T^{2} \)
67 \( 1 - 345. iT - 3.00e5T^{2} \)
71 \( 1 + 55.5T + 3.57e5T^{2} \)
73 \( 1 + 270.T + 3.89e5T^{2} \)
79 \( 1 - 927. iT - 4.93e5T^{2} \)
83 \( 1 - 173.T + 5.71e5T^{2} \)
89 \( 1 + 1.52e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.01e3T + 9.12e5T^{2} \)
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

−10.52870592620979674335305869858, −9.873451122962016988553120150954, −8.842187883317416020696863203989, −8.299723280554442209931452927457, −7.17176760476715671411062813977, −6.00950977204303543720871245361, −5.18241944393752457389668741327, −3.90236299892312798069535298713, −2.99685937503881008855115715230, −2.03739314090919220651111724379, 0.48441387503986938270279723283, 1.41977187789040477664222364096, 2.83941710712402962579118385168, 4.02591409733682003654580436251, 5.07947844828348747803112477001, 6.34818361087700563139706132137, 7.22010830406350887119993060105, 7.966876571571042428141554350647, 8.802332592287296444234470385360, 9.562060022951910250962358855153

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