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Properties

Label 2-252-63.5-c3-0-8
Degree $2$
Conductor $252$
Sign $0.950 - 0.310i$
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  + (3.03 − 4.21i)3-s + (5.20 + 9.01i)5-s + (−16.5 + 8.40i)7-s + (−8.51 − 25.6i)9-s + (46.6 + 26.9i)11-s + (25.4 + 14.6i)13-s + (53.7 + 5.46i)15-s + (39.8 + 69.0i)17-s + (25.7 + 14.8i)19-s + (−14.7 + 95.0i)21-s + (−0.928 + 0.535i)23-s + (8.35 − 14.4i)25-s + (−133. − 41.9i)27-s + (91.0 − 52.5i)29-s + 126. i·31-s + ⋯
L(s)  = 1  + (0.585 − 0.811i)3-s + (0.465 + 0.806i)5-s + (−0.891 + 0.453i)7-s + (−0.315 − 0.948i)9-s + (1.27 + 0.737i)11-s + (0.542 + 0.312i)13-s + (0.925 + 0.0941i)15-s + (0.568 + 0.984i)17-s + (0.310 + 0.179i)19-s + (−0.153 + 0.988i)21-s + (−0.00841 + 0.00485i)23-s + (0.0668 − 0.115i)25-s + (−0.954 − 0.299i)27-s + (0.582 − 0.336i)29-s + 0.730i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.950 - 0.310i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.950 - 0.310i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.315737554\)
\(L(\frac12)\) \(\approx\) \(2.315737554\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.03 + 4.21i)T \)
7 \( 1 + (16.5 - 8.40i)T \)
good5 \( 1 + (-5.20 - 9.01i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-46.6 - 26.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-25.4 - 14.6i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-39.8 - 69.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-25.7 - 14.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.928 - 0.535i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-91.0 + 52.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 126. iT - 2.97e4T^{2} \)
37 \( 1 + (-42.3 + 73.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-121. + 210. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-201. - 349. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 119.T + 1.03e5T^{2} \)
53 \( 1 + (119. - 68.7i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 695.T + 2.05e5T^{2} \)
61 \( 1 - 372. iT - 2.26e5T^{2} \)
67 \( 1 + 673.T + 3.00e5T^{2} \)
71 \( 1 + 139. iT - 3.57e5T^{2} \)
73 \( 1 + (-215. + 124. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.14e3T + 4.93e5T^{2} \)
83 \( 1 + (263. + 455. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-416. + 720. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.09e3 + 633. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.99937752605873907687489354642, −10.59495026539029359747653333876, −9.549044132957597011837503891517, −8.873338234288051521054313031444, −7.56289566775630651237483714511, −6.48485933817579582305499759118, −6.12050725232538686434338881760, −3.88485666505864977794353655077, −2.78114356443338679183264920961, −1.47828666500428134737533445933, 0.988398109335305825204957805029, 3.03101959128711827013278895618, 4.00794692524572088594648051187, 5.26988496782090938548199695654, 6.38743525951720911749277958248, 7.80991202554814909260181837936, 9.123362181361435128527972607343, 9.301198305403468636443590115425, 10.40275816541273903039561663918, 11.44871452723487609074893572540

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