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Properties

Label 2-252-28.27-c3-0-56
Degree $2$
Conductor $252$
Sign $-0.996 - 0.0891i$
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  + (1.79 − 2.18i)2-s + (−1.53 − 7.85i)4-s − 17.7i·5-s + (5.15 − 17.7i)7-s + (−19.8 − 10.7i)8-s + (−38.7 − 31.9i)10-s − 10.7i·11-s + 72.4i·13-s + (−29.5 − 43.2i)14-s + (−59.3 + 24.0i)16-s + 62.7i·17-s + 98.1·19-s + (−139. + 27.2i)20-s + (−23.5 − 19.3i)22-s − 160. i·23-s + ⋯
L(s)  = 1  + (0.635 − 0.771i)2-s + (−0.191 − 0.981i)4-s − 1.58i·5-s + (0.278 − 0.960i)7-s + (−0.879 − 0.476i)8-s + (−1.22 − 1.01i)10-s − 0.295i·11-s + 1.54i·13-s + (−0.564 − 0.825i)14-s + (−0.926 + 0.375i)16-s + 0.895i·17-s + 1.18·19-s + (−1.56 + 0.304i)20-s + (−0.228 − 0.187i)22-s − 1.45i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.996 - 0.0891i$
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: \((2,\ 252,\ (\ :3/2),\ -0.996 - 0.0891i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.162182158\)
\(L(\frac12)\) \(\approx\) \(2.162182158\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.79 + 2.18i)T \)
3 \( 1 \)
7 \( 1 + (-5.15 + 17.7i)T \)
good5 \( 1 + 17.7iT - 125T^{2} \)
11 \( 1 + 10.7iT - 1.33e3T^{2} \)
13 \( 1 - 72.4iT - 2.19e3T^{2} \)
17 \( 1 - 62.7iT - 4.91e3T^{2} \)
19 \( 1 - 98.1T + 6.85e3T^{2} \)
23 \( 1 + 160. iT - 1.21e4T^{2} \)
29 \( 1 - 90.1T + 2.43e4T^{2} \)
31 \( 1 + 201.T + 2.97e4T^{2} \)
37 \( 1 + 139.T + 5.06e4T^{2} \)
41 \( 1 + 297. iT - 6.89e4T^{2} \)
43 \( 1 - 22.8iT - 7.95e4T^{2} \)
47 \( 1 - 484.T + 1.03e5T^{2} \)
53 \( 1 - 502.T + 1.48e5T^{2} \)
59 \( 1 + 148.T + 2.05e5T^{2} \)
61 \( 1 + 438. iT - 2.26e5T^{2} \)
67 \( 1 + 667. iT - 3.00e5T^{2} \)
71 \( 1 + 174. iT - 3.57e5T^{2} \)
73 \( 1 - 1.16e3iT - 3.89e5T^{2} \)
79 \( 1 - 680. iT - 4.93e5T^{2} \)
83 \( 1 + 780.T + 5.71e5T^{2} \)
89 \( 1 - 27.2iT - 7.04e5T^{2} \)
97 \( 1 + 212. iT - 9.12e5T^{2} \)
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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

−11.27674175199418194620771026796, −10.30152640072019797713628136200, −9.224299096206668101617966422437, −8.512146143013193363986005664301, −6.96047057016855803403206866345, −5.57935451996435408258386874033, −4.55284657724475079885459779261, −3.87831687173600911937874018377, −1.79511167379903828997330764014, −0.71238201867175026072348015098, 2.67705866873337113186705933571, 3.39455904274150991734712406685, 5.23997877216267185651263446344, 5.90628265947233904559809695511, 7.23197956049264222912076364967, 7.68388318304669922766268541524, 9.073661648358963172222969230598, 10.21881875952886970997619992701, 11.40840431637019293536763273209, 12.02966783994212033836196292276

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