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Properties

Label 2-252-28.27-c3-0-29
Degree $2$
Conductor $252$
Sign $0.987 + 0.158i$
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 + (−51.8 + 7.73i)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 + (−0.989 + 0.147i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.987 + 0.158i$
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: \((2,\ 252,\ (\ :3/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.301877631\)
\(L(\frac12)\) \(\approx\) \(1.301877631\)
\(L(\frac{5}{2})\) 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

−11.16566013057068678856237524385, −10.70004061175567024744663684427, −9.874325963864224700198739508961, −8.532851989034594314941885402733, −7.78277119834981101510206373585, −6.90831155111010016945288657996, −5.63677937002913135574110500095, −3.76341882678142828492956033841, −2.58083611563614278548706164829, −0.967971998143645836277459062665, 1.03166803310029350256070513454, 2.28152200223224802437867139132, 4.57898553025406356539943137891, 5.49616525779422679030250040190, 6.86713271949094925335222178863, 7.86458990252480174475917628688, 8.827808124620847651450060403260, 9.366186736487281119230060346465, 10.53722452584947393162284320189, 11.73842114462744016176319799827

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