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Properties

Label 2-252-28.27-c3-0-26
Degree $2$
Conductor $252$
Sign $-0.565 - 0.824i$
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  + (2.41 + 1.47i)2-s + (3.65 + 7.11i)4-s + 17.0i·5-s + (18.3 − 2.33i)7-s + (−1.65 + 22.5i)8-s + (−25.1 + 41.2i)10-s − 41.4i·11-s + 45.3i·13-s + (47.7 + 21.4i)14-s + (−37.2 + 52.0i)16-s + 28.2i·17-s − 41.5·19-s + (−121. + 62.4i)20-s + (61.1 − 100. i)22-s − 93.9i·23-s + ⋯
L(s)  = 1  + (0.853 + 0.521i)2-s + (0.457 + 0.889i)4-s + 1.52i·5-s + (0.992 − 0.126i)7-s + (−0.0732 + 0.997i)8-s + (−0.795 + 1.30i)10-s − 1.13i·11-s + 0.967i·13-s + (0.912 + 0.409i)14-s + (−0.582 + 0.813i)16-s + 0.403i·17-s − 0.501·19-s + (−1.35 + 0.698i)20-s + (0.592 − 0.970i)22-s − 0.851i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.565 - 0.824i$
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: \((2,\ 252,\ (\ :3/2),\ -0.565 - 0.824i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.093579183\)
\(L(\frac12)\) \(\approx\) \(3.093579183\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.41 - 1.47i)T \)
3 \( 1 \)
7 \( 1 + (-18.3 + 2.33i)T \)
good5 \( 1 - 17.0iT - 125T^{2} \)
11 \( 1 + 41.4iT - 1.33e3T^{2} \)
13 \( 1 - 45.3iT - 2.19e3T^{2} \)
17 \( 1 - 28.2iT - 4.91e3T^{2} \)
19 \( 1 + 41.5T + 6.85e3T^{2} \)
23 \( 1 + 93.9iT - 1.21e4T^{2} \)
29 \( 1 + 27.8T + 2.43e4T^{2} \)
31 \( 1 - 81.4T + 2.97e4T^{2} \)
37 \( 1 - 94.8T + 5.06e4T^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 - 171. iT - 7.95e4T^{2} \)
47 \( 1 + 286.T + 1.03e5T^{2} \)
53 \( 1 - 575.T + 1.48e5T^{2} \)
59 \( 1 - 411.T + 2.05e5T^{2} \)
61 \( 1 + 778. iT - 2.26e5T^{2} \)
67 \( 1 - 198. iT - 3.00e5T^{2} \)
71 \( 1 + 197. iT - 3.57e5T^{2} \)
73 \( 1 - 255. iT - 3.89e5T^{2} \)
79 \( 1 + 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 - 938.T + 5.71e5T^{2} \)
89 \( 1 + 1.16e3iT - 7.04e5T^{2} \)
97 \( 1 + 656. iT - 9.12e5T^{2} \)
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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

−11.67803813440655402865197697390, −11.24548239753855783387565680411, −10.41673549171554960149363897286, −8.659123068200992760873419682024, −7.78820223330438970743273742278, −6.71112268405474115248878377887, −6.06410103932072928934116255570, −4.60677971329333379998433838659, −3.46679016476891576318566202243, −2.24839016956513153714692151370, 0.975610726858598407021592039130, 2.16621738089763585129377493763, 4.07195637779127095495422876586, 4.97211560485219882292485477837, 5.57340742538305531586719494530, 7.28620682641208574870037198580, 8.405905422153926261380122383575, 9.474861457328998854030654002826, 10.43813870648772830180512507628, 11.62534356650732749164314623017

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