↑

Properties

Label 2-504-63.16-c1-0-8
Degree $2$
Conductor $504$
Sign $0.296 - 0.954i$
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  + (1.5 + 0.866i)3-s − 5-s + (−0.5 + 2.59i)7-s + (1.5 + 2.59i)9-s + 3·11-s + (−1.5 − 2.59i)13-s + (−1.5 − 0.866i)15-s + (2.5 + 4.33i)17-s + (−3.5 + 6.06i)19-s + (−3 + 3.46i)21-s + 5·23-s − 4·25-s + 5.19i·27-s + (0.5 − 0.866i)29-s + (4 − 6.92i)31-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s − 0.447·5-s + (−0.188 + 0.981i)7-s + (0.5 + 0.866i)9-s + 0.904·11-s + (−0.416 − 0.720i)13-s + (−0.387 − 0.223i)15-s + (0.606 + 1.05i)17-s + (−0.802 + 1.39i)19-s + (−0.654 + 0.755i)21-s + 1.04·23-s − 0.800·25-s + 0.999i·27-s + (0.0928 − 0.160i)29-s + (0.718 − 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40066 + 1.03148i\)
\(L(\frac12)\) \(\approx\) \(1.40066 + 1.03148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5T + 23T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.5 + 4.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.99321326781109460624647963434, −10.00244110392701928421738923272, −9.368855721191488560239449223186, −8.273764678488830876456614724138, −7.949844659855592479942030744922, −6.46200027497635871010877652054, −5.43190040477714300381050085227, −4.12414223899831191220057553973, −3.30615592379222126337092899991, −1.98373169413292194729373076570, 1.03489778439806539903652537880, 2.71231913483851643270817127005, 3.83821401452245282404180715912, 4.73073569597464302296172720116, 6.66015960246056255851955522814, 7.03319985861831776750603198936, 7.934329716336804817417203836670, 9.080721053900358188440352486787, 9.533667934822457606964142978699, 10.78962526738704078092181979222

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