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Properties

Label 2-252-28.27-c3-0-35
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

\(L(2)\) \(\approx\) \(3.641729728\)
\(L(\frac12)\) \(\approx\) \(3.641729728\)
\(L(\frac{5}{2})\) 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

−11.58917687456205967878603796587, −10.99636644688252927052219291748, −10.07415817881660272039606558188, −8.696199841731614847840138960700, −7.38166236311015842616349529427, −6.43199801585546278178060590653, −5.39907244343737742787511410691, −4.18730648640643140947112981291, −2.93687167606777282471949190126, −1.68340038714252014356930161032, 1.27368779829303292789361852688, 3.01347098163786011665957278543, 4.66296386125366916920216729536, 4.98193793653516233638517572549, 6.38548245088418542501121200136, 7.71389174202939848999540481922, 8.224481020121293104805500374189, 9.701210483293787480381149449957, 10.92488404970047201395287940642, 11.78153778990683086794247799199

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