↑

Properties

Label 2-624-12.11-c3-0-42
Degree $2$
Conductor $624$
Sign $0.954 - 0.297i$
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  + (3.81 + 3.52i)3-s − 18.3i·5-s + 21.7i·7-s + (2.18 + 26.9i)9-s + 64.6·11-s − 13·13-s + (64.6 − 70.1i)15-s − 44.0i·17-s − 30.8i·19-s + (−76.7 + 83.1i)21-s + 85.6·23-s − 211.·25-s + (−86.4 + 110. i)27-s + 22.3i·29-s + 10.5i·31-s + ⋯
L(s)  = 1  + (0.735 + 0.677i)3-s − 1.64i·5-s + 1.17i·7-s + (0.0807 + 0.996i)9-s + 1.77·11-s − 0.277·13-s + (1.11 − 1.20i)15-s − 0.628i·17-s − 0.372i·19-s + (−0.797 + 0.864i)21-s + 0.776·23-s − 1.69·25-s + (−0.616 + 0.787i)27-s + 0.143i·29-s + 0.0613i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.954 - 0.297i$
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: \((2,\ 624,\ (\ :3/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.949615523\)
\(L(\frac12)\) \(\approx\) \(2.949615523\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.81 - 3.52i)T \)
13 \( 1 + 13T \)
good5 \( 1 + 18.3iT - 125T^{2} \)
7 \( 1 - 21.7iT - 343T^{2} \)
11 \( 1 - 64.6T + 1.33e3T^{2} \)
17 \( 1 + 44.0iT - 4.91e3T^{2} \)
19 \( 1 + 30.8iT - 6.85e3T^{2} \)
23 \( 1 - 85.6T + 1.21e4T^{2} \)
29 \( 1 - 22.3iT - 2.43e4T^{2} \)
31 \( 1 - 10.5iT - 2.97e4T^{2} \)
37 \( 1 - 237.T + 5.06e4T^{2} \)
41 \( 1 - 35.5iT - 6.89e4T^{2} \)
43 \( 1 - 499. iT - 7.95e4T^{2} \)
47 \( 1 + 204.T + 1.03e5T^{2} \)
53 \( 1 + 669. iT - 1.48e5T^{2} \)
59 \( 1 - 541.T + 2.05e5T^{2} \)
61 \( 1 - 929.T + 2.26e5T^{2} \)
67 \( 1 - 325. iT - 3.00e5T^{2} \)
71 \( 1 - 307.T + 3.57e5T^{2} \)
73 \( 1 - 752.T + 3.89e5T^{2} \)
79 \( 1 + 1.29e3iT - 4.93e5T^{2} \)
83 \( 1 - 945.T + 5.71e5T^{2} \)
89 \( 1 - 1.27e3iT - 7.04e5T^{2} \)
97 \( 1 + 550.T + 9.12e5T^{2} \)
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

−9.618579018934826003586827164149, −9.373982165737838246998220173124, −8.723641806960006407503920159683, −8.060788002733661112421507825382, −6.61339397906148251606596344516, −5.31429918342444675782067462574, −4.74046121353493097867489885534, −3.72488822726590339089211268457, −2.35135831127094153389794797475, −1.08654901844638948088925706074, 1.01271512624292967638980271201, 2.28193795944439274603704093679, 3.54237529857513489022102171945, 3.98922522621677361586455814106, 6.11200546867064423202084487179, 6.91070003637956589118017703295, 7.18255970003991982501442918355, 8.252562588083500255085848653711, 9.378876017087823061310725141374, 10.14583337037560467564355126476

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