Properties

Label 2-252-63.5-c3-0-1
Degree $2$
Conductor $252$
Sign $-0.989 + 0.144i$
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  + (3.58 + 3.76i)3-s + (−1.13 − 1.96i)5-s + (−14.5 + 11.4i)7-s + (−1.31 + 26.9i)9-s + (−31.0 − 17.9i)11-s + (−70.2 − 40.5i)13-s + (3.33 − 11.3i)15-s + (25.8 + 44.7i)17-s + (−9.61 − 5.55i)19-s + (−95.2 − 13.4i)21-s + (−26.4 + 15.2i)23-s + (59.9 − 103. i)25-s + (−106. + 91.7i)27-s + (−167. + 96.9i)29-s + 188. i·31-s + ⋯
L(s)  = 1  + (0.689 + 0.724i)3-s + (−0.101 − 0.176i)5-s + (−0.783 + 0.620i)7-s + (−0.0486 + 0.998i)9-s + (−0.851 − 0.491i)11-s + (−1.49 − 0.865i)13-s + (0.0574 − 0.195i)15-s + (0.368 + 0.638i)17-s + (−0.116 − 0.0670i)19-s + (−0.990 − 0.139i)21-s + (−0.239 + 0.138i)23-s + (0.479 − 0.830i)25-s + (−0.756 + 0.653i)27-s + (−1.07 + 0.620i)29-s + 1.08i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.989 + 0.144i$
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: \((2,\ 252,\ (\ :3/2),\ -0.989 + 0.144i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4942348630\)
\(L(\frac12)\) \(\approx\) \(0.4942348630\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.58 - 3.76i)T \)
7 \( 1 + (14.5 - 11.4i)T \)
good5 \( 1 + (1.13 + 1.96i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (31.0 + 17.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (70.2 + 40.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-25.8 - 44.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (9.61 + 5.55i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (26.4 - 15.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (167. - 96.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 188. iT - 2.97e4T^{2} \)
37 \( 1 + (-25.3 + 43.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-151. + 261. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-28.1 - 48.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 108.T + 1.03e5T^{2} \)
53 \( 1 + (548. - 316. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 686.T + 2.05e5T^{2} \)
61 \( 1 - 393. iT - 2.26e5T^{2} \)
67 \( 1 - 18.8T + 3.00e5T^{2} \)
71 \( 1 - 1.11e3iT - 3.57e5T^{2} \)
73 \( 1 + (-458. + 264. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 463.T + 4.93e5T^{2} \)
83 \( 1 + (-212. - 368. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-74.5 + 129. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.05e3 - 611. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.40994180286724643877601774944, −10.82132300099100212961744399083, −10.13180736060340157622341145648, −9.254896043349265262367476902425, −8.320110273707281364002056119642, −7.42087097687797033559731056368, −5.77572544291448943035548982567, −4.88126706635397882987614026370, −3.36870956644619304014895637955, −2.48370419645429470629425649459, 0.16271379764877894235597367044, 2.12532624402145526143682298924, 3.25621012556198163755313042985, 4.67864122177531730873934995474, 6.31601775194822210320863656631, 7.35601919017270090484381069592, 7.75313952624326479789471125685, 9.458056097507047337178102575689, 9.739189997216068132845783153484, 11.19885444522899016991289388167

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