Properties

Label 2-252-28.27-c3-0-0
Degree $2$
Conductor $252$
Sign $-0.478 - 0.878i$
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  + (−2.69 − 0.856i)2-s + (6.53 + 4.61i)4-s − 10.3i·5-s + (−16.6 − 8.15i)7-s + (−13.6 − 18.0i)8-s + (−8.83 + 27.7i)10-s − 18.0i·11-s + 49.0i·13-s + (37.8 + 36.2i)14-s + (21.3 + 60.3i)16-s − 46.6i·17-s − 48.8·19-s + (47.6 − 67.3i)20-s + (−15.4 + 48.6i)22-s + 33.7i·23-s + ⋯
L(s)  = 1  + (−0.952 − 0.302i)2-s + (0.816 + 0.577i)4-s − 0.921i·5-s + (−0.897 − 0.440i)7-s + (−0.603 − 0.797i)8-s + (−0.279 + 0.878i)10-s − 0.494i·11-s + 1.04i·13-s + (0.722 + 0.691i)14-s + (0.332 + 0.942i)16-s − 0.666i·17-s − 0.589·19-s + (0.532 − 0.752i)20-s + (−0.149 + 0.471i)22-s + 0.306i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07441720474\)
\(L(\frac12)\) \(\approx\) \(0.07441720474\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.69 + 0.856i)T \)
3 \( 1 \)
7 \( 1 + (16.6 + 8.15i)T \)
good5 \( 1 + 10.3iT - 125T^{2} \)
11 \( 1 + 18.0iT - 1.33e3T^{2} \)
13 \( 1 - 49.0iT - 2.19e3T^{2} \)
17 \( 1 + 46.6iT - 4.91e3T^{2} \)
19 \( 1 + 48.8T + 6.85e3T^{2} \)
23 \( 1 - 33.7iT - 1.21e4T^{2} \)
29 \( 1 + 48.1T + 2.43e4T^{2} \)
31 \( 1 + 152.T + 2.97e4T^{2} \)
37 \( 1 - 37.6T + 5.06e4T^{2} \)
41 \( 1 - 409. iT - 6.89e4T^{2} \)
43 \( 1 - 470. iT - 7.95e4T^{2} \)
47 \( 1 + 548.T + 1.03e5T^{2} \)
53 \( 1 - 203.T + 1.48e5T^{2} \)
59 \( 1 + 717.T + 2.05e5T^{2} \)
61 \( 1 - 493. iT - 2.26e5T^{2} \)
67 \( 1 + 240. iT - 3.00e5T^{2} \)
71 \( 1 - 995. iT - 3.57e5T^{2} \)
73 \( 1 + 790. iT - 3.89e5T^{2} \)
79 \( 1 + 214. iT - 4.93e5T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 + 67.3iT - 7.04e5T^{2} \)
97 \( 1 + 934. iT - 9.12e5T^{2} \)
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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

−11.73461479127970249755419389423, −10.94285894160112224621503840811, −9.697072091635274813004313537266, −9.211927561015032994150486917004, −8.238658348507665207227062797318, −7.08953080080408845597821909307, −6.13736468900251889080548316007, −4.43243955309962168699147288131, −3.07660890104141205424189159358, −1.37006523864020470581703312032, 0.04049908989939986391806504233, 2.18386791455983477343098432285, 3.37844139646684023526086716010, 5.52018501100598008295008638276, 6.48182985861608770478091579220, 7.26414325575004781438162617878, 8.389464399710484383746599584413, 9.387300475687373133524877863453, 10.37791716894575002590748134297, 10.80501527797336398510055813586

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