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Properties

Label 2-132-44.19-c1-0-6
Degree $2$
Conductor $132$
Sign $0.635 + 0.772i$
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  + (−1.12 + 0.861i)2-s + (−0.951 + 0.309i)3-s + (0.514 − 1.93i)4-s + (−1.53 − 1.11i)5-s + (0.799 − 1.16i)6-s + (1.07 − 3.29i)7-s + (1.08 + 2.61i)8-s + (0.809 − 0.587i)9-s + (2.67 − 0.0726i)10-s + (0.210 − 3.30i)11-s + (0.108 + 1.99i)12-s + (1.06 + 1.46i)13-s + (1.64 + 4.62i)14-s + (1.80 + 0.585i)15-s + (−3.47 − 1.98i)16-s + (1.74 − 2.40i)17-s + ⋯
L(s)  = 1  + (−0.792 + 0.609i)2-s + (−0.549 + 0.178i)3-s + (0.257 − 0.966i)4-s + (−0.685 − 0.497i)5-s + (0.326 − 0.476i)6-s + (0.405 − 1.24i)7-s + (0.385 + 0.922i)8-s + (0.269 − 0.195i)9-s + (0.846 − 0.0229i)10-s + (0.0636 − 0.997i)11-s + (0.0312 + 0.576i)12-s + (0.296 + 0.407i)13-s + (0.438 + 1.23i)14-s + (0.465 + 0.151i)15-s + (−0.867 − 0.496i)16-s + (0.424 − 0.583i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486687 - 0.229832i\)
\(L(\frac12)\) \(\approx\) \(0.486687 - 0.229832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.861i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-0.210 + 3.30i)T \)
good5 \( 1 + (1.53 + 1.11i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.07 + 3.29i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.06 - 1.46i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.74 + 2.40i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.34 + 4.14i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.82iT - 23T^{2} \)
29 \( 1 + (8.48 + 2.75i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.65 - 2.27i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.51 - 10.8i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-7.37 + 2.39i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.35T + 43T^{2} \)
47 \( 1 + (-2.66 + 0.866i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.05 + 3.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.68 - 2.49i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.57 + 6.29i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 15.0iT - 67T^{2} \)
71 \( 1 + (-3.24 + 4.46i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.1 - 3.62i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.34 - 3.15i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.10 + 4.43i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.20T + 89T^{2} \)
97 \( 1 + (-6.72 + 4.88i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.36769421781422098072124559124, −11.58579736712568584161904308981, −11.13254949572629737528309684095, −10.00715501266362053972076752358, −8.797628802020221354959457023795, −7.76270990080741751354875598656, −6.81397745692282807627641744015, −5.39135210990585656988026126524, −4.11640516094926808386232627784, −0.808207834784076840775235610836, 2.11274947746771700785732281042, 3.90564964004336714125364794821, 5.74593340399183380996667445581, 7.25203306084649151005715613742, 8.135514033400838257430922969490, 9.311239619227830170621255994571, 10.55288701441970901030109799760, 11.31071278954694898111293926626, 12.32510567240169799923370603854, 12.69245620176164115676687395758

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