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Properties

Label 2-252-63.5-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.890 - 0.455i$
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  + (−4.76 − 2.06i)3-s + (9.23 + 16.0i)5-s + (−13.5 − 12.6i)7-s + (18.4 + 19.6i)9-s + (43.7 + 25.2i)11-s + (−68.1 − 39.3i)13-s + (−11.0 − 95.3i)15-s + (2.61 + 4.52i)17-s + (−23.3 − 13.5i)19-s + (38.4 + 88.2i)21-s + (−50.4 + 29.1i)23-s + (−108. + 187. i)25-s + (−47.4 − 132. i)27-s + (−133. + 76.9i)29-s + 2.06i·31-s + ⋯
L(s)  = 1  + (−0.917 − 0.397i)3-s + (0.826 + 1.43i)5-s + (−0.730 − 0.682i)7-s + (0.684 + 0.729i)9-s + (1.20 + 0.693i)11-s + (−1.45 − 0.839i)13-s + (−0.189 − 1.64i)15-s + (0.0372 + 0.0645i)17-s + (−0.282 − 0.163i)19-s + (0.399 + 0.916i)21-s + (−0.457 + 0.263i)23-s + (−0.865 + 1.49i)25-s + (−0.338 − 0.941i)27-s + (−0.853 + 0.492i)29-s + 0.0119i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.890 - 0.455i$
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: \((2,\ 252,\ (\ :3/2),\ -0.890 - 0.455i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4891837270\)
\(L(\frac12)\) \(\approx\) \(0.4891837270\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.76 + 2.06i)T \)
7 \( 1 + (13.5 + 12.6i)T \)
good5 \( 1 + (-9.23 - 16.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-43.7 - 25.2i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (68.1 + 39.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-2.61 - 4.52i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (23.3 + 13.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (50.4 - 29.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (133. - 76.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 2.06iT - 2.97e4T^{2} \)
37 \( 1 + (184. - 319. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (199. - 345. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (152. + 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 294.T + 1.03e5T^{2} \)
53 \( 1 + (431. - 249. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 157.T + 2.05e5T^{2} \)
61 \( 1 + 726. iT - 2.26e5T^{2} \)
67 \( 1 + 164.T + 3.00e5T^{2} \)
71 \( 1 - 157. iT - 3.57e5T^{2} \)
73 \( 1 + (-908. + 524. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 198.T + 4.93e5T^{2} \)
83 \( 1 + (492. + 853. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-45.0 + 78.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (785. - 453. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.98077361266719451079952997325, −10.95947881723153572055867066388, −10.02099716935856414702353397600, −9.758974516024329956875127962767, −7.57581655375874329053011294087, −6.79560513624622659446942585712, −6.31378338349778862652810119881, −4.94364635393204395817300888775, −3.30832192322372824311376082125, −1.81896493439178063606724610460, 0.20737276934907600963077743882, 1.83279792523176071480036688486, 3.99039616968401544860782016473, 5.11842179457227542825209589861, 5.89917409437591552459331145383, 6.81860820294265809454688789544, 8.697482120659344983569150013009, 9.430107819591028824946065209069, 9.891015496670821669799971233576, 11.43408692331315392799343072402

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