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Properties

Label 2-252-63.38-c3-0-18
Degree $2$
Conductor $252$
Sign $-0.502 + 0.864i$
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  + (−3.44 − 3.88i)3-s + (−0.950 + 1.64i)5-s + (14.2 − 11.7i)7-s + (−3.23 + 26.8i)9-s + (33.6 − 19.4i)11-s + (−8.40 + 4.85i)13-s + (9.67 − 1.97i)15-s + (22.9 − 39.6i)17-s + (19.7 − 11.4i)19-s + (−95.0 − 14.9i)21-s + (−135. − 78.4i)23-s + (60.6 + 105. i)25-s + (115. − 79.8i)27-s + (−187. − 108. i)29-s − 201. i·31-s + ⋯
L(s)  = 1  + (−0.663 − 0.748i)3-s + (−0.0849 + 0.147i)5-s + (0.771 − 0.635i)7-s + (−0.119 + 0.992i)9-s + (0.923 − 0.533i)11-s + (−0.179 + 0.103i)13-s + (0.166 − 0.0340i)15-s + (0.326 − 0.565i)17-s + (0.239 − 0.138i)19-s + (−0.987 − 0.155i)21-s + (−1.23 − 0.711i)23-s + (0.485 + 0.841i)25-s + (0.822 − 0.568i)27-s + (−1.19 − 0.692i)29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.502 + 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.246135137\)
\(L(\frac12)\) \(\approx\) \(1.246135137\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.44 + 3.88i)T \)
7 \( 1 + (-14.2 + 11.7i)T \)
good5 \( 1 + (0.950 - 1.64i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-33.6 + 19.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (8.40 - 4.85i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-22.9 + 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-19.7 + 11.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (135. + 78.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (187. + 108. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 201. iT - 2.97e4T^{2} \)
37 \( 1 + (146. + 253. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (119. + 206. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (158. - 275. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 152.T + 1.03e5T^{2} \)
53 \( 1 + (26.3 + 15.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 466.T + 2.05e5T^{2} \)
61 \( 1 + 506. iT - 2.26e5T^{2} \)
67 \( 1 - 630.T + 3.00e5T^{2} \)
71 \( 1 - 413. iT - 3.57e5T^{2} \)
73 \( 1 + (-276. - 159. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (-415. + 719. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (572. + 990. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.55e3 - 899. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.40807070613993717225363912117, −10.64393605463145812310512227640, −9.364906953502681175878210503622, −8.039782546307781137529710064826, −7.31254407564874189852904689118, −6.28288018086843651305451516496, −5.18099938077577441609612805962, −3.88058296780488399041959130717, −1.94608294189378776506603109158, −0.56468015000349558988444206191, 1.56841457956488543098678936231, 3.58971228169605403641698259255, 4.73640497888129770102907403279, 5.62867332373555221512485890503, 6.75748912230023285427708548027, 8.175757791396139954078343237780, 9.139750721848385756523104356626, 10.05240833966008462262776717664, 10.98953549674400449697099492454, 12.11795172616094059592480943800

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