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Properties

Label 2-252-63.41-c3-0-11
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: $\chi_{252} (41, \cdot )$
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

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

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