Properties

Label 2-252-84.11-c1-0-2
Degree $2$
Conductor $252$
Sign $-0.167 - 0.985i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 + 0.179i)2-s + (1.93 − 0.504i)4-s + (−0.604 + 0.349i)5-s + (−1.16 + 2.37i)7-s + (−2.62 + 1.05i)8-s + (0.785 − 0.598i)10-s + (−1.27 + 2.21i)11-s + 1.88·13-s + (1.20 − 3.54i)14-s + (3.49 − 1.95i)16-s + (3.44 + 1.98i)17-s + (−6.11 + 3.52i)19-s + (−0.994 + 0.980i)20-s + (1.39 − 3.32i)22-s + (2.01 + 3.48i)23-s + ⋯
L(s)  = 1  + (−0.991 + 0.127i)2-s + (0.967 − 0.252i)4-s + (−0.270 + 0.156i)5-s + (−0.439 + 0.898i)7-s + (−0.927 + 0.373i)8-s + (0.248 − 0.189i)10-s + (−0.384 + 0.666i)11-s + 0.521·13-s + (0.321 − 0.946i)14-s + (0.872 − 0.488i)16-s + (0.834 + 0.481i)17-s + (−1.40 + 0.809i)19-s + (−0.222 + 0.219i)20-s + (0.296 − 0.709i)22-s + (0.419 + 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396781 + 0.470003i\)
\(L(\frac12)\) \(\approx\) \(0.396781 + 0.470003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.179i)T \)
3 \( 1 \)
7 \( 1 + (1.16 - 2.37i)T \)
good5 \( 1 + (0.604 - 0.349i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.27 - 2.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.88T + 13T^{2} \)
17 \( 1 + (-3.44 - 1.98i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.11 - 3.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.01 - 3.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.86iT - 29T^{2} \)
31 \( 1 + (0.815 + 0.470i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.74 - 6.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + 3.97iT - 43T^{2} \)
47 \( 1 + (-4.45 - 7.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.458 - 0.264i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.65 + 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.18 + 8.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.35 - 1.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.51T + 71T^{2} \)
73 \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.7 + 6.76i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + (10.2 - 5.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−12.21349596902644080454948187606, −11.17612501112004232724177742642, −10.27525406744202766993891740728, −9.396439570510052059109682064744, −8.450707883903971085562678820681, −7.58508766580279954719118184779, −6.41508462676600122855205294586, −5.48922695655088644480463930554, −3.47771405487670726215288052653, −1.96482005015997039355290657830, 0.65984234961584076839895294726, 2.77591427591482991186268516994, 4.15115061784637937732837971006, 6.00590669832060208380283313275, 6.98799857531266709199084654393, 8.009584076891334526326606070249, 8.810714296689942007960955606311, 9.951138505548750755936539047083, 10.71961812940206998833932090689, 11.45827824736829473069508604680

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