Properties

Label 2-252-63.20-c3-0-14
Degree $2$
Conductor $252$
Sign $-0.187 + 0.982i$
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  + (−4.66 − 2.29i)3-s + (−5.16 − 8.94i)5-s + (14.4 + 11.5i)7-s + (16.4 + 21.3i)9-s + (27.1 + 15.6i)11-s + (39.0 − 22.5i)13-s + (3.57 + 53.5i)15-s − 62.4·17-s − 132. i·19-s + (−41.1 − 86.9i)21-s + (−58.8 + 33.9i)23-s + (9.14 − 15.8i)25-s + (−27.8 − 137. i)27-s + (−116. − 67.3i)29-s + (25.9 − 15.0i)31-s + ⋯
L(s)  = 1  + (−0.897 − 0.441i)3-s + (−0.461 − 0.800i)5-s + (0.782 + 0.622i)7-s + (0.610 + 0.791i)9-s + (0.743 + 0.429i)11-s + (0.832 − 0.480i)13-s + (0.0616 + 0.921i)15-s − 0.891·17-s − 1.60i·19-s + (−0.428 − 0.903i)21-s + (−0.533 + 0.307i)23-s + (0.0731 − 0.126i)25-s + (−0.198 − 0.980i)27-s + (−0.747 − 0.431i)29-s + (0.150 − 0.0869i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.187 + 0.982i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.139631167\)
\(L(\frac12)\) \(\approx\) \(1.139631167\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.66 + 2.29i)T \)
7 \( 1 + (-14.4 - 11.5i)T \)
good5 \( 1 + (5.16 + 8.94i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.1 - 15.6i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-39.0 + 22.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 62.4T + 4.91e3T^{2} \)
19 \( 1 + 132. iT - 6.85e3T^{2} \)
23 \( 1 + (58.8 - 33.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (116. + 67.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-25.9 + 15.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 40.4T + 5.06e4T^{2} \)
41 \( 1 + (39.7 + 68.8i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-161. + 279. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-171. + 296. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 64.9iT - 1.48e5T^{2} \)
59 \( 1 + (79.3 + 137. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (493. + 285. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-150. - 261. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 719. iT - 3.57e5T^{2} \)
73 \( 1 - 558. iT - 3.89e5T^{2} \)
79 \( 1 + (-456. + 790. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-352. + 610. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 700.T + 7.04e5T^{2} \)
97 \( 1 + (-202. - 117. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.49456947411706834630514967018, −10.74695166430119442732479351221, −9.205160534401851650094253882299, −8.416062134330626369505768227903, −7.32719063342222864898675120635, −6.18408616493005608313814676487, −5.09283338015685056043931602662, −4.24431089783476898768839033693, −1.97397163792564986236529579602, −0.57268994883659572784482335795, 1.34689005690750957041447709361, 3.66272804635094171745634110463, 4.36641670794985176137599933526, 5.90155160618577079927541080266, 6.72451194305531149288544940678, 7.81457656616289663719158168584, 9.054476308935793905034230508065, 10.30005177736889008327454412318, 11.10373922621359914914778302352, 11.44346073318422401978776657284

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