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Properties

Label 2-252-28.27-c3-0-21
Degree $2$
Conductor $252$
Sign $0.997 + 0.0716i$
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  + (0.951 − 2.66i)2-s + (−6.19 − 5.06i)4-s + 10.8i·5-s + (15.1 − 10.6i)7-s + (−19.3 + 11.6i)8-s + (28.8 + 10.3i)10-s + 45.0i·11-s + 40.3i·13-s + (−14.0 − 50.4i)14-s + (12.6 + 62.7i)16-s − 8.11i·17-s + 53.3·19-s + (54.9 − 67.1i)20-s + (119. + 42.8i)22-s + 55.7i·23-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)2-s + (−0.773 − 0.633i)4-s + 0.970i·5-s + (0.817 − 0.576i)7-s + (−0.856 + 0.515i)8-s + (0.913 + 0.326i)10-s + 1.23i·11-s + 0.861i·13-s + (−0.267 − 0.963i)14-s + (0.197 + 0.980i)16-s − 0.115i·17-s + 0.643·19-s + (0.614 − 0.750i)20-s + (1.16 + 0.415i)22-s + 0.505i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0716i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.997 + 0.0716i$
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: \((2,\ 252,\ (\ :3/2),\ 0.997 + 0.0716i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.985158583\)
\(L(\frac12)\) \(\approx\) \(1.985158583\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 2.66i)T \)
3 \( 1 \)
7 \( 1 + (-15.1 + 10.6i)T \)
good5 \( 1 - 10.8iT - 125T^{2} \)
11 \( 1 - 45.0iT - 1.33e3T^{2} \)
13 \( 1 - 40.3iT - 2.19e3T^{2} \)
17 \( 1 + 8.11iT - 4.91e3T^{2} \)
19 \( 1 - 53.3T + 6.85e3T^{2} \)
23 \( 1 - 55.7iT - 1.21e4T^{2} \)
29 \( 1 - 169.T + 2.43e4T^{2} \)
31 \( 1 - 262.T + 2.97e4T^{2} \)
37 \( 1 + 354.T + 5.06e4T^{2} \)
41 \( 1 + 42.7iT - 6.89e4T^{2} \)
43 \( 1 - 23.1iT - 7.95e4T^{2} \)
47 \( 1 - 437.T + 1.03e5T^{2} \)
53 \( 1 + 388.T + 1.48e5T^{2} \)
59 \( 1 + 649.T + 2.05e5T^{2} \)
61 \( 1 - 916. iT - 2.26e5T^{2} \)
67 \( 1 - 736. iT - 3.00e5T^{2} \)
71 \( 1 + 600. iT - 3.57e5T^{2} \)
73 \( 1 - 673. iT - 3.89e5T^{2} \)
79 \( 1 + 585. iT - 4.93e5T^{2} \)
83 \( 1 - 856.T + 5.71e5T^{2} \)
89 \( 1 + 920. iT - 7.04e5T^{2} \)
97 \( 1 - 638. iT - 9.12e5T^{2} \)
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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

−11.67375450785873672869413255792, −10.60257987015181577166272589011, −10.09434516486338129939516749305, −8.949403917655956347430536519936, −7.55511140272130480708585546294, −6.59552732761750352765625967070, −5.00752986802708382069913099091, −4.12640081823797318837975654958, −2.73006281014872327301528088915, −1.44781911923498986616077046617, 0.802729191435037445129949315607, 3.12446081884937395040709496629, 4.69843426968415162054040708976, 5.37806520758929650797581248794, 6.37457669126523761732827949219, 7.976675509927166341783461778869, 8.398450545118342466200299056448, 9.233863877086400904059428789787, 10.68940976871231827881133262129, 12.00687256689619240551187251269

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