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Properties

Label 2-252-28.27-c3-0-10
Degree $2$
Conductor $252$
Sign $-0.519 - 0.854i$
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  + (−0.414 + 2.79i)2-s + (−7.65 − 2.31i)4-s − 8.74i·5-s + (−13.8 − 12.3i)7-s + (9.65 − 20.4i)8-s + (24.4 + 3.62i)10-s + 0.397i·11-s + 75.7i·13-s + (40.2 − 33.4i)14-s + (53.2 + 35.4i)16-s + 84.4i·17-s + 20.0·19-s + (−20.2 + 66.9i)20-s + (−1.11 − 0.164i)22-s + 122. i·23-s + ⋯
L(s)  = 1  + (−0.146 + 0.989i)2-s + (−0.957 − 0.289i)4-s − 0.782i·5-s + (−0.745 − 0.666i)7-s + (0.426 − 0.904i)8-s + (0.773 + 0.114i)10-s + 0.0109i·11-s + 1.61i·13-s + (0.768 − 0.639i)14-s + (0.832 + 0.554i)16-s + 1.20i·17-s + 0.242·19-s + (−0.226 + 0.748i)20-s + (−0.0107 − 0.00159i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.519 - 0.854i$
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: \((2,\ 252,\ (\ :3/2),\ -0.519 - 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.020160294\)
\(L(\frac12)\) \(\approx\) \(1.020160294\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.414 - 2.79i)T \)
3 \( 1 \)
7 \( 1 + (13.8 + 12.3i)T \)
good5 \( 1 + 8.74iT - 125T^{2} \)
11 \( 1 - 0.397iT - 1.33e3T^{2} \)
13 \( 1 - 75.7iT - 2.19e3T^{2} \)
17 \( 1 - 84.4iT - 4.91e3T^{2} \)
19 \( 1 - 20.0T + 6.85e3T^{2} \)
23 \( 1 - 122. iT - 1.21e4T^{2} \)
29 \( 1 - 175.T + 2.43e4T^{2} \)
31 \( 1 + 128.T + 2.97e4T^{2} \)
37 \( 1 - 253.T + 5.06e4T^{2} \)
41 \( 1 - 81.4iT - 6.89e4T^{2} \)
43 \( 1 + 69.1iT - 7.95e4T^{2} \)
47 \( 1 - 147.T + 1.03e5T^{2} \)
53 \( 1 + 283.T + 1.48e5T^{2} \)
59 \( 1 + 632.T + 2.05e5T^{2} \)
61 \( 1 - 3.25iT - 2.26e5T^{2} \)
67 \( 1 - 551. iT - 3.00e5T^{2} \)
71 \( 1 + 486. iT - 3.57e5T^{2} \)
73 \( 1 - 165. iT - 3.89e5T^{2} \)
79 \( 1 - 279. iT - 4.93e5T^{2} \)
83 \( 1 - 622.T + 5.71e5T^{2} \)
89 \( 1 - 696. iT - 7.04e5T^{2} \)
97 \( 1 - 1.62e3iT - 9.12e5T^{2} \)
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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

−12.17823230410806252477840970595, −10.76318480695355258497993865976, −9.608304938169377126183280429581, −9.044074389801328591620605665531, −7.944062025704278679215674814919, −6.88915496543217349459998875721, −6.05010910718309676323554715708, −4.70502370528893507951738886305, −3.80765250014432627294164915546, −1.24084448970809512818125974727, 0.48990779076566087795949744242, 2.65975018653638702929702114511, 3.17435395130982011735776171050, 4.87983717727416780062713648079, 6.09166446244299667084121204862, 7.45926836797596625531971045565, 8.592262420898629994713513056576, 9.603792035913856867572470914366, 10.39084392636513060329895433580, 11.15838049921616950251079897906

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