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Properties

Label 2-252-63.20-c3-0-3
Degree $2$
Conductor $252$
Sign $-0.866 - 0.499i$
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  + (−0.0939 + 5.19i)3-s + (−7.82 − 13.5i)5-s + (17.6 + 5.69i)7-s + (−26.9 − 0.976i)9-s + (34.2 + 19.7i)11-s + (−55.5 + 32.0i)13-s + (71.1 − 39.3i)15-s − 56.6·17-s + 117. i·19-s + (−31.2 + 91.0i)21-s + (−6.59 + 3.80i)23-s + (−59.9 + 103. i)25-s + (7.61 − 140. i)27-s + (39.8 + 22.9i)29-s + (−251. + 145. i)31-s + ⋯
L(s)  = 1  + (−0.0180 + 0.999i)3-s + (−0.699 − 1.21i)5-s + (0.951 + 0.307i)7-s + (−0.999 − 0.0361i)9-s + (0.937 + 0.541i)11-s + (−1.18 + 0.684i)13-s + (1.22 − 0.677i)15-s − 0.808·17-s + 1.41i·19-s + (−0.324 + 0.945i)21-s + (−0.0597 + 0.0345i)23-s + (−0.479 + 0.830i)25-s + (0.0542 − 0.998i)27-s + (0.254 + 0.147i)29-s + (−1.45 + 0.842i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.866 - 0.499i$
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: \((2,\ 252,\ (\ :3/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8388473148\)
\(L(\frac12)\) \(\approx\) \(0.8388473148\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.0939 - 5.19i)T \)
7 \( 1 + (-17.6 - 5.69i)T \)
good5 \( 1 + (7.82 + 13.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-34.2 - 19.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (55.5 - 32.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 56.6T + 4.91e3T^{2} \)
19 \( 1 - 117. iT - 6.85e3T^{2} \)
23 \( 1 + (6.59 - 3.80i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-39.8 - 22.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (251. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 + (97.2 + 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (318. - 550. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 274. iT - 1.48e5T^{2} \)
59 \( 1 + (-258. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (142. + 82.1i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-368. - 637. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 599. iT - 3.57e5T^{2} \)
73 \( 1 - 214. iT - 3.89e5T^{2} \)
79 \( 1 + (454. - 786. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-389. + 675. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 443.T + 7.04e5T^{2} \)
97 \( 1 + (337. + 194. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.03087263705918090707198599734, −11.18209125928974428377620533373, −9.961129178187092770525234827003, −8.964303728580934482409587802312, −8.478746539519050299269480533235, −7.19306469099824884144841875389, −5.44238305611079895638755308890, −4.62736077599956382540843051837, −3.91724396936898969156372613849, −1.78134471406512378655301413150, 0.32315610590414050690994258894, 2.16157171640329378441201629974, 3.41567308810965334955310305407, 4.98720061026271581850407241986, 6.52258723181846063646057404025, 7.20909373233592991013254373563, 7.946274032730471111074802796817, 9.058578406915813341996553943043, 10.66365708871162214410310642301, 11.33877529153766235334256774129

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