↑

Properties

Label 2-504-56.53-c1-0-33
Degree $2$
Conductor $504$
Sign $-0.308 + 0.951i$
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  + (1.00 − 0.990i)2-s + (0.0372 − 1.99i)4-s + (−0.586 + 0.338i)5-s + (2.23 − 1.41i)7-s + (−1.94 − 2.05i)8-s + (−0.256 + 0.922i)10-s + (−1.44 − 0.835i)11-s − 1.28i·13-s + (0.857 − 3.64i)14-s + (−3.99 − 0.148i)16-s + (3.18 − 5.51i)17-s + (2.20 − 1.27i)19-s + (0.655 + 1.18i)20-s + (−2.28 + 0.590i)22-s + (−0.127 − 0.221i)23-s + ⋯
L(s)  = 1  + (0.713 − 0.700i)2-s + (0.0186 − 0.999i)4-s + (−0.262 + 0.151i)5-s + (0.845 − 0.534i)7-s + (−0.687 − 0.726i)8-s + (−0.0811 + 0.291i)10-s + (−0.436 − 0.251i)11-s − 0.355i·13-s + (0.229 − 0.973i)14-s + (−0.999 − 0.0372i)16-s + (0.771 − 1.33i)17-s + (0.506 − 0.292i)19-s + (0.146 + 0.265i)20-s + (−0.487 + 0.125i)22-s + (−0.0266 − 0.0461i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.308 + 0.951i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18243 - 1.62600i\)
\(L(\frac12)\) \(\approx\) \(1.18243 - 1.62600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.00 + 0.990i)T \)
3 \( 1 \)
7 \( 1 + (-2.23 + 1.41i)T \)
good5 \( 1 + (0.586 - 0.338i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.44 + 0.835i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.28iT - 13T^{2} \)
17 \( 1 + (-3.18 + 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.127 + 0.221i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.27iT - 29T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.62 - 3.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 + 5.48iT - 43T^{2} \)
47 \( 1 + (-4.73 - 8.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.91 + 2.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.74 - 5.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.2 - 7.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.6 - 7.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (1.43 - 2.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.09 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.63iT - 83T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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

−10.87502782736567599840311630264, −10.04931887236118040312438150377, −9.088567578184104471194224596740, −7.79084304442091319118011012340, −7.03789221392398293309352089688, −5.55885060668257784760278819741, −4.93449195616920302517578634836, −3.73730355541507260441419869108, −2.68957804639082598595473279712, −1.04667332721590517964197927988, 2.12353793411361916099450122575, 3.63231502241912161018852683386, 4.64584532314035201370957059499, 5.55438584071391571393475252787, 6.41366348608167385638145320937, 7.80862603457931092335634845224, 8.062802230858992813637253353806, 9.145082919273298867477834115440, 10.39936879951532850695739108578, 11.48135928233785392783283055957

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