Properties

Label 2-252-63.47-c3-0-12
Degree $2$
Conductor $252$
Sign $0.811 - 0.583i$
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  + (4.93 + 1.63i)3-s − 19.9·5-s + (16.1 − 9.13i)7-s + (21.6 + 16.0i)9-s + 5.52i·11-s + (37.0 − 21.4i)13-s + (−98.5 − 32.6i)15-s + (37.3 + 64.6i)17-s + (128. + 74.1i)19-s + (94.3 − 18.8i)21-s + 108. i·23-s + 274.·25-s + (80.6 + 114. i)27-s + (70.0 + 40.4i)29-s + (−212. − 122. i)31-s + ⋯
L(s)  = 1  + (0.949 + 0.313i)3-s − 1.78·5-s + (0.869 − 0.493i)7-s + (0.802 + 0.596i)9-s + 0.151i·11-s + (0.791 − 0.456i)13-s + (−1.69 − 0.561i)15-s + (0.532 + 0.922i)17-s + (1.55 + 0.895i)19-s + (0.980 − 0.195i)21-s + 0.987i·23-s + 2.19·25-s + (0.575 + 0.818i)27-s + (0.448 + 0.258i)29-s + (−1.23 − 0.711i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.811 - 0.583i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.140913705\)
\(L(\frac12)\) \(\approx\) \(2.140913705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.93 - 1.63i)T \)
7 \( 1 + (-16.1 + 9.13i)T \)
good5 \( 1 + 19.9T + 125T^{2} \)
11 \( 1 - 5.52iT - 1.33e3T^{2} \)
13 \( 1 + (-37.0 + 21.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-37.3 - 64.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-128. - 74.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 108. iT - 1.21e4T^{2} \)
29 \( 1 + (-70.0 - 40.4i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (212. + 122. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-42.6 + 73.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-12.0 - 20.8i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (208. - 361. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (87.8 + 152. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-512. + 296. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-275. + 476. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (308. - 178. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-363. + 629. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 486. iT - 3.57e5T^{2} \)
73 \( 1 + (76.4 - 44.1i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-557. - 965. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-395. + 684. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (224. - 389. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (610. + 352. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−11.55958352752010315759841050355, −10.86561858037948036511833724631, −9.765846200044805847484046910662, −8.369720731908264346880522736565, −7.936951685460868530873042089300, −7.27106440739873235236192480756, −5.22028560127038951663305250574, −3.88883194811915066089131018838, −3.48417120240815156345579931438, −1.29881731330960755871813973888, 0.966501303681985485032384901010, 2.85140720945868204163577142690, 3.89561444186155880603487961242, 5.02190543536406491681382286705, 6.97683909942360526979970954536, 7.65590224512428477667959266974, 8.509233758693272257545217093192, 9.141468344626657128036469122109, 10.77504247148046489399282482559, 11.79941115835341382088918730848

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