Properties

Label 2-252-63.47-c3-0-9
Degree $2$
Conductor $252$
Sign $0.449 - 0.893i$
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  + (−4.40 + 2.75i)3-s + 3.45·5-s + (13.4 + 12.7i)7-s + (11.7 − 24.2i)9-s − 31.0i·11-s + (2.16 − 1.24i)13-s + (−15.2 + 9.53i)15-s + (−0.621 − 1.07i)17-s + (76.0 + 43.8i)19-s + (−94.3 − 19.1i)21-s + 49.7i·23-s − 113.·25-s + (15.1 + 139. i)27-s + (228. + 132. i)29-s + (222. + 128. i)31-s + ⋯
L(s)  = 1  + (−0.847 + 0.530i)3-s + 0.309·5-s + (0.724 + 0.689i)7-s + (0.436 − 0.899i)9-s − 0.852i·11-s + (0.0461 − 0.0266i)13-s + (−0.262 + 0.164i)15-s + (−0.00886 − 0.0153i)17-s + (0.918 + 0.530i)19-s + (−0.979 − 0.199i)21-s + 0.450i·23-s − 0.904·25-s + (0.107 + 0.994i)27-s + (1.46 + 0.845i)29-s + (1.28 + 0.744i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.449 - 0.893i$
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: \((2,\ 252,\ (\ :3/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.503397181\)
\(L(\frac12)\) \(\approx\) \(1.503397181\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.40 - 2.75i)T \)
7 \( 1 + (-13.4 - 12.7i)T \)
good5 \( 1 - 3.45T + 125T^{2} \)
11 \( 1 + 31.0iT - 1.33e3T^{2} \)
13 \( 1 + (-2.16 + 1.24i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (0.621 + 1.07i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-76.0 - 43.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 49.7iT - 1.21e4T^{2} \)
29 \( 1 + (-228. - 132. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-222. - 128. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (47.7 - 82.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (93.9 + 162. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (133. - 230. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-285. - 495. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (167. - 96.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-145. + 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (137. - 79.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-221. + 384. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 405. iT - 3.57e5T^{2} \)
73 \( 1 + (505. - 291. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-643. - 1.11e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-610. + 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-406. + 704. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (496. + 286. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−11.75677396221574453074136064402, −10.87859444985719924120339584575, −9.974596243465114267463642735459, −8.978726718098508569376348625827, −7.942297181421155668361171705122, −6.43103988183860143327430352612, −5.58643706023879970675887007289, −4.71345011378679791980918030115, −3.17834429639396213804334982499, −1.23564339109715215779328967638, 0.799726652714271992805563149725, 2.17305731784555730828220384034, 4.32003167047127254774919197409, 5.21738280212492291862379871345, 6.46738127327164459628293379605, 7.35034058453076214945076169491, 8.244385845470222737316203375780, 9.842901155761704464730270272321, 10.47766660531450501990313718510, 11.63834803014440659778045247960

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