↑

Properties

Label 2-252-28.27-c3-0-47
Degree $2$
Conductor $252$
Sign $-0.791 + 0.611i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
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  + (−2.82 + 0.0488i)2-s + (7.99 − 0.276i)4-s − 16.6i·5-s + (15.0 − 10.8i)7-s + (−22.5 + 1.17i)8-s + (0.813 + 47.0i)10-s − 64.0i·11-s − 28.6i·13-s + (−42.0 + 31.2i)14-s + (63.8 − 4.41i)16-s + 82.9i·17-s − 17.1·19-s + (−4.60 − 133. i)20-s + (3.12 + 181. i)22-s + 95.0i·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0172i)2-s + (0.999 − 0.0345i)4-s − 1.48i·5-s + (0.812 − 0.583i)7-s + (−0.998 + 0.0517i)8-s + (0.0257 + 1.48i)10-s − 1.75i·11-s − 0.612i·13-s + (−0.801 + 0.597i)14-s + (0.997 − 0.0690i)16-s + 1.18i·17-s − 0.206·19-s + (−0.0514 − 1.48i)20-s + (0.0303 + 1.75i)22-s + 0.861i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.791 + 0.611i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.791 + 0.611i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.007112639\)
\(L(\frac12)\) \(\approx\) \(1.007112639\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 - 0.0488i)T \)
3 \( 1 \)
7 \( 1 + (-15.0 + 10.8i)T \)
good5 \( 1 + 16.6iT - 125T^{2} \)
11 \( 1 + 64.0iT - 1.33e3T^{2} \)
13 \( 1 + 28.6iT - 2.19e3T^{2} \)
17 \( 1 - 82.9iT - 4.91e3T^{2} \)
19 \( 1 + 17.1T + 6.85e3T^{2} \)
23 \( 1 - 95.0iT - 1.21e4T^{2} \)
29 \( 1 - 197.T + 2.43e4T^{2} \)
31 \( 1 + 153.T + 2.97e4T^{2} \)
37 \( 1 - 10.7T + 5.06e4T^{2} \)
41 \( 1 - 41.1iT - 6.89e4T^{2} \)
43 \( 1 + 412. iT - 7.95e4T^{2} \)
47 \( 1 + 477.T + 1.03e5T^{2} \)
53 \( 1 - 35.2T + 1.48e5T^{2} \)
59 \( 1 + 494.T + 2.05e5T^{2} \)
61 \( 1 - 294. iT - 2.26e5T^{2} \)
67 \( 1 + 207. iT - 3.00e5T^{2} \)
71 \( 1 + 534. iT - 3.57e5T^{2} \)
73 \( 1 - 582. iT - 3.89e5T^{2} \)
79 \( 1 + 311. iT - 4.93e5T^{2} \)
83 \( 1 - 1.31e3T + 5.71e5T^{2} \)
89 \( 1 + 616. iT - 7.04e5T^{2} \)
97 \( 1 + 104. iT - 9.12e5T^{2} \)
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

−11.05437752260746729503624948419, −10.30408336554377967751298231735, −8.957594097751712798334965798623, −8.403028047973269987663620476670, −7.76456046814169455342753226996, −6.13853131766824257900704382554, −5.16932801669212707887704566785, −3.58415659686293822948163038815, −1.52775793385551043345620892241, −0.56566836568425442112774076990, 1.92107999675924466189613051776, 2.80989386872649746892693862084, 4.74333411625759574107563713137, 6.43991515350137217841702914742, 7.10311915810163052341713818718, 7.962489059173455321234938691585, 9.253780708693849961310523743512, 10.03134840845219010685189266600, 10.94119711787898085449219737660, 11.66574390922703943422237947089

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