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Properties

Label 2-252-63.58-c3-0-14
Degree $2$
Conductor $252$
Sign $0.991 - 0.131i$
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  + (4.87 − 1.81i)3-s + (10.8 + 18.7i)5-s + (−12.6 − 13.5i)7-s + (20.4 − 17.6i)9-s + (29.2 − 50.7i)11-s + (4.45 − 7.71i)13-s + (86.7 + 71.7i)15-s + (49.1 + 85.1i)17-s + (32.3 − 56.0i)19-s + (−86.1 − 42.8i)21-s + (72.8 + 126. i)23-s + (−172. + 298. i)25-s + (67.5 − 122. i)27-s + (93.8 + 162. i)29-s − 2.85·31-s + ⋯
L(s)  = 1  + (0.937 − 0.348i)3-s + (0.968 + 1.67i)5-s + (−0.683 − 0.729i)7-s + (0.756 − 0.653i)9-s + (0.802 − 1.38i)11-s + (0.0950 − 0.164i)13-s + (1.49 + 1.23i)15-s + (0.701 + 1.21i)17-s + (0.390 − 0.676i)19-s + (−0.895 − 0.445i)21-s + (0.660 + 1.14i)23-s + (−1.37 + 2.38i)25-s + (0.481 − 0.876i)27-s + (0.601 + 1.04i)29-s − 0.0165·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.991 - 0.131i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.991 - 0.131i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.955014429\)
\(L(\frac12)\) \(\approx\) \(2.955014429\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.87 + 1.81i)T \)
7 \( 1 + (12.6 + 13.5i)T \)
good5 \( 1 + (-10.8 - 18.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-29.2 + 50.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-4.45 + 7.71i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-49.1 - 85.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-32.3 + 56.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-72.8 - 126. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-93.8 - 162. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 2.85T + 2.97e4T^{2} \)
37 \( 1 + (6.08 - 10.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-74.1 + 128. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (76.8 + 133. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 370.T + 1.03e5T^{2} \)
53 \( 1 + (218. + 378. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 416.T + 2.05e5T^{2} \)
61 \( 1 - 302.T + 2.26e5T^{2} \)
67 \( 1 + 339.T + 3.00e5T^{2} \)
71 \( 1 + 544.T + 3.57e5T^{2} \)
73 \( 1 + (-119. - 206. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 39.2T + 4.93e5T^{2} \)
83 \( 1 + (578. + 1.00e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-764. + 1.32e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (817. + 1.41e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.39532653550355847973884006258, −10.50726698684469204879289566387, −9.776000348842029170638244859295, −8.813736875402897963107590775194, −7.45297099442270332308027608190, −6.68369496777944839079356304385, −5.93970399004627837515733347165, −3.41342511163739673993588721062, −3.20228458402347161058834533023, −1.44460669781025436764790118858, 1.37933525193208584187872327255, 2.61318097535398400843552370558, 4.35847359751365226145554284646, 5.15358455984330329570602485594, 6.48280141846453215975978246789, 7.954585426181056279927999470077, 8.987167145114560728612292360433, 9.555142039087983798909382354031, 9.951850290323140376268153113339, 12.05437499773095267886652984778

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