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Properties

Label 2-132-44.35-c1-0-4
Degree $2$
Conductor $132$
Sign $-0.159 - 0.987i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + (0.574 + 1.29i)2-s + (0.587 + 0.809i)3-s + (−1.34 + 1.48i)4-s + (0.0244 + 0.0751i)5-s + (−0.707 + 1.22i)6-s + (0.500 + 0.363i)7-s + (−2.68 − 0.878i)8-s + (−0.309 + 0.951i)9-s + (−0.0830 + 0.0747i)10-s + (0.0857 − 3.31i)11-s + (−1.98 − 0.211i)12-s + (3.41 + 1.10i)13-s + (−0.182 + 0.856i)14-s + (−0.0464 + 0.0639i)15-s + (−0.408 − 3.97i)16-s + (−2.15 + 0.700i)17-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.339 + 0.467i)3-s + (−0.670 + 0.742i)4-s + (0.0109 + 0.0336i)5-s + (−0.288 + 0.499i)6-s + (0.189 + 0.137i)7-s + (−0.950 − 0.310i)8-s + (−0.103 + 0.317i)9-s + (−0.0262 + 0.0236i)10-s + (0.0258 − 0.999i)11-s + (−0.574 − 0.0610i)12-s + (0.946 + 0.307i)13-s + (−0.0487 + 0.228i)14-s + (−0.0119 + 0.0165i)15-s + (−0.102 − 0.994i)16-s + (−0.522 + 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.876272 + 1.02944i\)
\(L(\frac12)\) \(\approx\) \(0.876272 + 1.02944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.574 - 1.29i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.0857 + 3.31i)T \)
good5 \( 1 + (-0.0244 - 0.0751i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.500 - 0.363i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-3.41 - 1.10i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.15 - 0.700i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.03 + 0.750i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.24iT - 23T^{2} \)
29 \( 1 + (4.38 - 6.03i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.81 - 0.914i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.85 - 1.34i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.06 + 9.72i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.23T + 43T^{2} \)
47 \( 1 + (-4.07 - 5.61i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.742 - 2.28i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.18 - 1.63i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.53 - 3.09i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + (9.14 - 2.97i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.36 - 7.38i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.38 - 7.33i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.80 + 5.54i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3.55T + 89T^{2} \)
97 \( 1 + (-4.20 + 12.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.80560892033306660084316539906, −12.90519658089659202911539979963, −11.58301541039693958175402443188, −10.44860894796212925680791620581, −8.791394020021637250352374679340, −8.541627023674560782103589705382, −6.94604081640382749269036099841, −5.81634139943070354847898459500, −4.51680898664916180765296939025, −3.22476068207846771046592261483, 1.68708346419943529303624799047, 3.35895493224157831147197956480, 4.78165415410411891592861808238, 6.23999275240678187525153099215, 7.72604172988947997263239464842, 9.011045124055733126137149484669, 9.966713142710883519765609552980, 11.17468402114644496763861188219, 11.98745706964177148211805504384, 13.17414717553662332433432074517

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