↑

Properties

Label 2-504-56.19-c1-0-10
Degree $2$
Conductor $504$
Sign $-0.530 - 0.847i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.138 + 1.40i)2-s + (−1.96 + 0.390i)4-s + (0.245 − 0.424i)5-s + (2.09 + 1.61i)7-s + (−0.821 − 2.70i)8-s + (0.632 + 0.286i)10-s + (1.81 + 3.14i)11-s + 0.540·13-s + (−1.98 + 3.17i)14-s + (3.69 − 1.53i)16-s + (−5.14 + 2.96i)17-s + (5.91 + 3.41i)19-s + (−0.315 + 0.929i)20-s + (−4.17 + 2.99i)22-s + (−5.00 − 2.88i)23-s + ⋯
L(s)  = 1  + (0.0981 + 0.995i)2-s + (−0.980 + 0.195i)4-s + (0.109 − 0.190i)5-s + (0.792 + 0.610i)7-s + (−0.290 − 0.956i)8-s + (0.199 + 0.0905i)10-s + (0.547 + 0.948i)11-s + 0.149·13-s + (−0.529 + 0.848i)14-s + (0.923 − 0.383i)16-s + (−1.24 + 0.720i)17-s + (1.35 + 0.783i)19-s + (−0.0704 + 0.207i)20-s + (−0.890 + 0.637i)22-s + (−1.04 − 0.602i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688924 + 1.24331i\)
\(L(\frac12)\) \(\approx\) \(0.688924 + 1.24331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.138 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-2.09 - 1.61i)T \)
good5 \( 1 + (-0.245 + 0.424i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.81 - 3.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.540T + 13T^{2} \)
17 \( 1 + (5.14 - 2.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.91 - 3.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.00 + 2.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.49iT - 29T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.156 + 0.0904i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.01iT - 41T^{2} \)
43 \( 1 + 2.87T + 43T^{2} \)
47 \( 1 + (-3.88 + 6.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.36 + 0.788i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 2.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.31 + 9.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.72 - 6.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.94iT - 71T^{2} \)
73 \( 1 + (-3.28 + 1.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.76 - 2.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.61iT - 83T^{2} \)
89 \( 1 + (8.57 + 4.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.12iT - 97T^{2} \)
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

−11.31789294924729568422047649438, −10.06956560250578725442814576093, −9.178970141067055878619639075461, −8.489554120848859167704855323915, −7.58063599117897680401060132844, −6.65396668267224370535728992262, −5.62061106855134405550777978734, −4.79571968236905258296089388112, −3.76401487256186930149818650632, −1.77732856703407154056009335046, 0.913330168102013257080159336524, 2.42566429961561160643746121641, 3.69931920183734442124407850580, 4.64336672842728287223852596702, 5.70132227323781831383844472671, 7.00823936519618381471098101253, 8.170910769835783006734137997014, 8.993888548346830332758275136824, 9.865988050640722374005256157724, 10.86083647862720516418759442117

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