Properties

Label 2-252-63.47-c3-0-5
Degree $2$
Conductor $252$
Sign $0.703 - 0.710i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 5.06i)3-s − 7.54·5-s + (−17.0 − 7.27i)7-s + (−24.3 + 11.7i)9-s + 56.7i·11-s + (41.0 − 23.6i)13-s + (8.71 + 38.2i)15-s + (23.2 + 40.3i)17-s + (85.5 + 49.3i)19-s + (−17.1 + 94.6i)21-s − 198. i·23-s − 68.0·25-s + (87.4 + 109. i)27-s + (151. + 87.4i)29-s + (−81.6 − 47.1i)31-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s − 0.674·5-s + (−0.919 − 0.392i)7-s + (−0.901 + 0.433i)9-s + 1.55i·11-s + (0.874 − 0.505i)13-s + (0.150 + 0.657i)15-s + (0.332 + 0.575i)17-s + (1.03 + 0.596i)19-s + (−0.178 + 0.983i)21-s − 1.79i·23-s − 0.544·25-s + (0.623 + 0.782i)27-s + (0.969 + 0.559i)29-s + (−0.473 − 0.273i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.703 - 0.710i$
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: \((2,\ 252,\ (\ :3/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8671406639\)
\(L(\frac12)\) \(\approx\) \(0.8671406639\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.15 + 5.06i)T \)
7 \( 1 + (17.0 + 7.27i)T \)
good5 \( 1 + 7.54T + 125T^{2} \)
11 \( 1 - 56.7iT - 1.33e3T^{2} \)
13 \( 1 + (-41.0 + 23.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-23.2 - 40.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-85.5 - 49.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 198. iT - 1.21e4T^{2} \)
29 \( 1 + (-151. - 87.4i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (81.6 + 47.1i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (216. - 375. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-129. - 224. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-177. + 307. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-234. - 406. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-83.4 + 48.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (244. - 424. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (604. - 348. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-6.70 + 11.6i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 787. iT - 3.57e5T^{2} \)
73 \( 1 + (725. - 418. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-236. - 409. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-216. + 375. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (611. - 1.05e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-174. - 100. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.19414485629761241142799305903, −10.76495892901588316860617527559, −9.972324367902865063025762093412, −8.562655343757814303134043391149, −7.62139675428173373989101853072, −6.84878958279774607661796103279, −5.85585222730276401598427482123, −4.29486711178037354291425833643, −2.93099054537304896812083768319, −1.18395340693035085930792585809, 0.41125454943334115576911795025, 3.19872974068086008426798396637, 3.75800486372218412977521368852, 5.38865503322915731756596335629, 6.14218601556707581498311870806, 7.60072022643915996074047176540, 8.933984179769325609377813752154, 9.354799375946313526103459703930, 10.65706488155133994265367045100, 11.46741891547656418122558209774

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