↑

Properties

Label 2-504-56.13-c2-0-53
Degree $2$
Conductor $504$
Sign $0.962 + 0.270i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 − 0.835i)2-s + (2.60 − 3.03i)4-s + 7.31·5-s + (0.0227 + 6.99i)7-s + (2.19 − 7.69i)8-s + (13.2 − 6.11i)10-s + 15.8i·11-s + 12.2·13-s + (5.89 + 12.7i)14-s + (−2.44 − 15.8i)16-s + 8.41i·17-s + 8.46·19-s + (19.0 − 22.2i)20-s + (13.2 + 28.8i)22-s − 45.5·23-s + ⋯
L(s)  = 1  + (0.908 − 0.417i)2-s + (0.650 − 0.759i)4-s + 1.46·5-s + (0.00324 + 0.999i)7-s + (0.273 − 0.961i)8-s + (1.32 − 0.611i)10-s + 1.44i·11-s + 0.946·13-s + (0.420 + 0.907i)14-s + (−0.152 − 0.988i)16-s + 0.494i·17-s + 0.445·19-s + (0.951 − 1.11i)20-s + (0.603 + 1.31i)22-s − 1.98·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ 0.962 + 0.270i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.967583421\)
\(L(\frac12)\) \(\approx\) \(3.967583421\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.81 + 0.835i)T \)
3 \( 1 \)
7 \( 1 + (-0.0227 - 6.99i)T \)
good5 \( 1 - 7.31T + 25T^{2} \)
11 \( 1 - 15.8iT - 121T^{2} \)
13 \( 1 - 12.2T + 169T^{2} \)
17 \( 1 - 8.41iT - 289T^{2} \)
19 \( 1 - 8.46T + 361T^{2} \)
23 \( 1 + 45.5T + 529T^{2} \)
29 \( 1 + 15.5iT - 841T^{2} \)
31 \( 1 + 41.0iT - 961T^{2} \)
37 \( 1 - 11.2iT - 1.36e3T^{2} \)
41 \( 1 + 22.8iT - 1.68e3T^{2} \)
43 \( 1 + 46.2iT - 1.84e3T^{2} \)
47 \( 1 - 62.8iT - 2.20e3T^{2} \)
53 \( 1 + 79.7iT - 2.80e3T^{2} \)
59 \( 1 + 27.2T + 3.48e3T^{2} \)
61 \( 1 - 15.7T + 3.72e3T^{2} \)
67 \( 1 + 55.8iT - 4.48e3T^{2} \)
71 \( 1 - 37.3T + 5.04e3T^{2} \)
73 \( 1 + 105. iT - 5.32e3T^{2} \)
79 \( 1 - 92.1T + 6.24e3T^{2} \)
83 \( 1 + 38.3T + 6.88e3T^{2} \)
89 \( 1 - 90.6iT - 7.92e3T^{2} \)
97 \( 1 - 26.7iT - 9.40e3T^{2} \)
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

−10.63889116068405764131209901863, −9.805529955963968904060746132536, −9.364862982228517724195606629384, −7.911489447885922727496070072436, −6.42777871672381566351431874822, −5.96134281810306018709167993165, −5.10272244497047193617712896690, −3.88640905768510783125216588140, −2.29705103596120955504121597846, −1.80710105389544623695703006680, 1.42935788973214040645683825275, 2.96544037888835078609127435863, 3.97360851110628365041854786031, 5.32236871108868899429682025564, 6.03349377837621038398114945635, 6.72077748012489532660080841581, 7.933743060087390333275435386985, 8.832463738957732179100370479003, 10.05050681375577378417434421923, 10.80078718338484425521857668181

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