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Properties

Label 2-252-63.5-c3-0-18
Degree $2$
Conductor $252$
Sign $-0.350 + 0.936i$
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  + (4.19 − 3.06i)3-s + (4.37 + 7.58i)5-s + (−7.70 − 16.8i)7-s + (8.26 − 25.7i)9-s + (−56.1 − 32.4i)11-s + (−29.8 − 17.2i)13-s + (41.6 + 18.4i)15-s + (−41.3 − 71.6i)17-s + (121. + 69.9i)19-s + (−83.8 − 47.1i)21-s + (115. − 66.7i)23-s + (24.1 − 41.8i)25-s + (−43.9 − 133. i)27-s + (−7.57 + 4.37i)29-s + 56.7i·31-s + ⋯
L(s)  = 1  + (0.808 − 0.589i)3-s + (0.391 + 0.678i)5-s + (−0.415 − 0.909i)7-s + (0.306 − 0.951i)9-s + (−1.53 − 0.888i)11-s + (−0.637 − 0.368i)13-s + (0.716 + 0.317i)15-s + (−0.590 − 1.02i)17-s + (1.46 + 0.844i)19-s + (−0.871 − 0.489i)21-s + (1.04 − 0.605i)23-s + (0.193 − 0.334i)25-s + (−0.313 − 0.949i)27-s + (−0.0484 + 0.0279i)29-s + 0.328i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.350 + 0.936i$
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: \((2,\ 252,\ (\ :3/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.846464798\)
\(L(\frac12)\) \(\approx\) \(1.846464798\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.19 + 3.06i)T \)
7 \( 1 + (7.70 + 16.8i)T \)
good5 \( 1 + (-4.37 - 7.58i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (56.1 + 32.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (29.8 + 17.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (41.3 + 71.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-121. - 69.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-115. + 66.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (7.57 - 4.37i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 56.7iT - 2.97e4T^{2} \)
37 \( 1 + (-130. + 225. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (237. - 411. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (116. + 201. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 32.8T + 1.03e5T^{2} \)
53 \( 1 + (218. - 126. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 145.T + 2.05e5T^{2} \)
61 \( 1 + 613. iT - 2.26e5T^{2} \)
67 \( 1 - 522.T + 3.00e5T^{2} \)
71 \( 1 - 447. iT - 3.57e5T^{2} \)
73 \( 1 + (386. - 223. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.39e3T + 4.93e5T^{2} \)
83 \( 1 + (-449. - 778. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-409. + 709. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (152. - 88.2i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.16521325372535510292266864796, −10.25303213215401257683848735845, −9.495517645636800275952335023509, −8.135100873175888923638453843646, −7.38697262695228413622316738717, −6.54591764778133494493155313774, −5.13319553880138932873505430645, −3.28991932646792535442785367506, −2.62507295699683548278856186614, −0.63360748813948583296319970989, 2.03477042220884270121998180115, 3.07593959848359346175900425587, 4.82575627623695469018786370358, 5.34516126222823803132068029106, 7.12049317702208496953563660901, 8.176412829103961659050199241426, 9.218465151737010393154690072992, 9.653656708118428898352597152707, 10.71519973215844976370041731346, 12.02431102262134874035822384876

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