Properties

Label 2-252-63.20-c3-0-8
Degree $2$
Conductor $252$
Sign $-0.00752 - 0.999i$
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  + (−3.66 + 3.68i)3-s + (8.29 + 14.3i)5-s + (18.2 − 3.26i)7-s + (−0.127 − 26.9i)9-s + (46.2 + 26.7i)11-s + (11.0 − 6.37i)13-s + (−83.3 − 22.1i)15-s + 96.7·17-s − 54.6i·19-s + (−54.7 + 79.1i)21-s + (−55.6 + 32.1i)23-s + (−75.2 + 130. i)25-s + (99.9 + 98.5i)27-s + (−112. − 65.0i)29-s + (−190. + 110. i)31-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)3-s + (0.742 + 1.28i)5-s + (0.984 − 0.176i)7-s + (−0.00470 − 0.999i)9-s + (1.26 + 0.732i)11-s + (0.235 − 0.136i)13-s + (−1.43 − 0.380i)15-s + 1.38·17-s − 0.660i·19-s + (−0.569 + 0.822i)21-s + (−0.504 + 0.291i)23-s + (−0.601 + 1.04i)25-s + (0.712 + 0.702i)27-s + (−0.721 − 0.416i)29-s + (−1.10 + 0.637i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00752 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00752 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.00752 - 0.999i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.00752 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.948444006\)
\(L(\frac12)\) \(\approx\) \(1.948444006\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.66 - 3.68i)T \)
7 \( 1 + (-18.2 + 3.26i)T \)
good5 \( 1 + (-8.29 - 14.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-46.2 - 26.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-11.0 + 6.37i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 96.7T + 4.91e3T^{2} \)
19 \( 1 + 54.6iT - 6.85e3T^{2} \)
23 \( 1 + (55.6 - 32.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (112. + 65.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (190. - 110. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 279.T + 5.06e4T^{2} \)
41 \( 1 + (-185. - 320. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (153. - 266. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (163. - 283. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 451. iT - 1.48e5T^{2} \)
59 \( 1 + (258. + 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (234. + 135. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (370. + 642. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 914. iT - 3.57e5T^{2} \)
73 \( 1 - 337. iT - 3.89e5T^{2} \)
79 \( 1 + (498. - 863. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (17.0 - 29.4i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 208.T + 7.04e5T^{2} \)
97 \( 1 + (1.10e3 + 635. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.40390465159288643697480616515, −11.05775759636641677157878525458, −9.911528632979879436766550627514, −9.430172855165562636537912447198, −7.73260387792179724719270982890, −6.64204653982290673300766831741, −5.80348902803805156188960567271, −4.57697581966275481408473480020, −3.33305794496782911854717504164, −1.53940847832826708974301499879, 1.00494023476183953793632332979, 1.75801891914165725095384220138, 4.16834211901894769869899292946, 5.56010476760519687975192653848, 5.85128949052282692955924158408, 7.45161242152376014618505373796, 8.454395774222029523134995691531, 9.256865494646707848048175035269, 10.55059821184077238919626852764, 11.68449624366177282848611560085

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