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Properties

Label 2-252-28.27-c3-0-52
Degree $2$
Conductor $252$
Sign $-0.718 - 0.695i$
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  + (−0.965 − 2.65i)2-s + (−6.13 + 5.13i)4-s − 19.4i·5-s + (−1.95 − 18.4i)7-s + (19.5 + 11.3i)8-s + (−51.7 + 18.7i)10-s − 24.6i·11-s + 5.25i·13-s + (−47.0 + 22.9i)14-s + (11.3 − 62.9i)16-s − 80.8i·17-s + 86.5·19-s + (99.8 + 119. i)20-s + (−65.5 + 23.7i)22-s + 108. i·23-s + ⋯
L(s)  = 1  + (−0.341 − 0.939i)2-s + (−0.767 + 0.641i)4-s − 1.74i·5-s + (−0.105 − 0.994i)7-s + (0.864 + 0.502i)8-s + (−1.63 + 0.593i)10-s − 0.675i·11-s + 0.112i·13-s + (−0.898 + 0.438i)14-s + (0.177 − 0.984i)16-s − 1.15i·17-s + 1.04·19-s + (1.11 + 1.33i)20-s + (−0.635 + 0.230i)22-s + 0.982i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.718 - 0.695i$
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: \((2,\ 252,\ (\ :3/2),\ -0.718 - 0.695i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9810533457\)
\(L(\frac12)\) \(\approx\) \(0.9810533457\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 2.65i)T \)
3 \( 1 \)
7 \( 1 + (1.95 + 18.4i)T \)
good5 \( 1 + 19.4iT - 125T^{2} \)
11 \( 1 + 24.6iT - 1.33e3T^{2} \)
13 \( 1 - 5.25iT - 2.19e3T^{2} \)
17 \( 1 + 80.8iT - 4.91e3T^{2} \)
19 \( 1 - 86.5T + 6.85e3T^{2} \)
23 \( 1 - 108. iT - 1.21e4T^{2} \)
29 \( 1 + 278.T + 2.43e4T^{2} \)
31 \( 1 - 116.T + 2.97e4T^{2} \)
37 \( 1 - 53.5T + 5.06e4T^{2} \)
41 \( 1 - 303. iT - 6.89e4T^{2} \)
43 \( 1 - 176. iT - 7.95e4T^{2} \)
47 \( 1 + 102.T + 1.03e5T^{2} \)
53 \( 1 + 185.T + 1.48e5T^{2} \)
59 \( 1 - 732.T + 2.05e5T^{2} \)
61 \( 1 + 443. iT - 2.26e5T^{2} \)
67 \( 1 + 166. iT - 3.00e5T^{2} \)
71 \( 1 - 378. iT - 3.57e5T^{2} \)
73 \( 1 - 664. iT - 3.89e5T^{2} \)
79 \( 1 + 737. iT - 4.93e5T^{2} \)
83 \( 1 - 913.T + 5.71e5T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.44e3iT - 9.12e5T^{2} \)
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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

−11.27926422848825964533170790560, −9.740430343916513467380053142795, −9.389483081933286381010293959475, −8.254398800671533191172608446922, −7.44756628702225930965639294913, −5.42512342873333761502307355435, −4.51941824970536157127654811275, −3.38716166031413923399414980531, −1.39397182131806343565641030015, −0.47194425263115504690713220283, 2.21541839125018790084324446465, 3.73604179811918556424423376726, 5.43928364244572928112127547834, 6.33948497488903246006572308161, 7.15540941594636959257041193614, 8.088218212195158326242057031233, 9.304863034977374266997076215386, 10.17095187269717001599933874194, 10.96138910630882655500520387068, 12.17022112128399063486033266796

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