Properties

Label 2-252-63.20-c3-0-12
Degree $2$
Conductor $252$
Sign $0.585 + 0.810i$
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 + (−13.7 − 12.4i)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 + (−65.7 + 70.2i)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 + (−0.742 − 0.670i)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 + (−0.683 + 0.729i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.585 + 0.810i$
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: \((2,\ 252,\ (\ :3/2),\ 0.585 + 0.810i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.991844531\)
\(L(\frac12)\) \(\approx\) \(1.991844531\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.7 + 12.4i)T \)
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

−11.42949354404242142808278455840, −10.53344633203889140549752479042, −9.672748240947536967081730973407, −8.429233515591395512992484636393, −7.03406507542917463976230699298, −6.71164540192656243160083823178, −5.72637910174862157656123768863, −3.62864617617223024109316581193, −2.55132803825081315613509465595, −0.944544047639874773211591326262, 1.32091497683759881937181155010, 3.28481464498201837883419467446, 4.38503574477893565549627266200, 5.73806477016757158384851135120, 6.16553452204609286477334526064, 8.378270075380311331814311122940, 8.988279845845769928483914243475, 9.612683264154797361020303027628, 10.59775570840520834852896987422, 11.91255611033329211201685174698

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