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Properties

Label 2-132-44.7-c1-0-1
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: $\chi_{132} (7, \cdot )$
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

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

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