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Properties

Label 2-132-132.119-c1-0-8
Degree $2$
Conductor $132$
Sign $0.825 + 0.564i$
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  + (−0.642 − 1.26i)2-s + (1.45 + 0.945i)3-s + (−1.17 + 1.61i)4-s + (−0.587 + 0.190i)5-s + (0.260 − 2.43i)6-s + (2.48 − 3.42i)7-s + (2.79 + 0.442i)8-s + (1.21 + 2.74i)9-s + (0.618 + 0.618i)10-s + (3.30 + 0.224i)11-s + (−3.23 + 1.23i)12-s + (−0.309 + 0.951i)13-s + (−5.91 − 0.937i)14-s + (−1.03 − 0.278i)15-s + (−1.23 − 3.80i)16-s + (−5.79 + 1.88i)17-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (0.837 + 0.546i)3-s + (−0.587 + 0.809i)4-s + (−0.262 + 0.0854i)5-s + (0.106 − 0.994i)6-s + (0.941 − 1.29i)7-s + (0.987 + 0.156i)8-s + (0.403 + 0.914i)9-s + (0.195 + 0.195i)10-s + (0.997 + 0.0676i)11-s + (−0.934 + 0.356i)12-s + (−0.0857 + 0.263i)13-s + (−1.58 − 0.250i)14-s + (−0.266 − 0.0719i)15-s + (−0.309 − 0.951i)16-s + (−1.40 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05514 - 0.326163i\)
\(L(\frac12)\) \(\approx\) \(1.05514 - 0.326163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 1.26i)T \)
3 \( 1 + (-1.45 - 0.945i)T \)
11 \( 1 + (-3.30 - 0.224i)T \)
good5 \( 1 + (0.587 - 0.190i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.48 + 3.42i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (5.79 - 1.88i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.08 + 1.5i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + (-2.35 + 3.23i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.812 - 0.263i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.66 + 3.38i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.310 - 0.427i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 + (-3.54 + 2.57i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.89 - 1.59i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.73 - 3.44i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.57 - 4.84i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.85iT - 67T^{2} \)
71 \( 1 + (1.57 + 4.84i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.61 + 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.92 + 1.60i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.16 + 12.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (3.80 - 11.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.42680682303613666418716030269, −11.85345856091081806332504118259, −10.96511505133950724318606732417, −10.18506288653908659457057974800, −9.060074949300420455899638612037, −8.155035827808414588733386444067, −7.16106256934328881894181036517, −4.38371688471194507700249783006, −3.93062461296592049058577059156, −1.92404432459410892658176311232, 1.98054709366043475606211930123, 4.33013277206191206055036986726, 5.90116877152067596527976677874, 7.02720204486259152616156842035, 8.376904320696485455583754198714, 8.635819596867457890386363429034, 9.794332978087963998164101653749, 11.48690028998872344778747315873, 12.41396070525717426296484480772, 13.80608364603462954709488865191

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