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Properties

Label 2-132-11.3-c1-0-1
Degree $2$
Conductor $132$
Sign $0.0694 + 0.997i$
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  + (−0.309 − 0.951i)3-s + (−2.30 − 1.67i)5-s + (1.30 − 4.02i)7-s + (−0.809 + 0.587i)9-s + (2.19 + 2.48i)11-s + (1.42 − 1.03i)13-s + (−0.881 + 2.71i)15-s + (−3.73 − 2.71i)17-s + (1.88 + 5.79i)19-s − 4.23·21-s + 4.23·23-s + (0.972 + 2.99i)25-s + (0.809 + 0.587i)27-s + (−1.38 + 4.25i)29-s + (6.97 − 5.06i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (−1.03 − 0.750i)5-s + (0.494 − 1.52i)7-s + (−0.269 + 0.195i)9-s + (0.660 + 0.750i)11-s + (0.395 − 0.287i)13-s + (−0.227 + 0.700i)15-s + (−0.906 − 0.658i)17-s + (0.431 + 1.32i)19-s − 0.924·21-s + 0.883·23-s + (0.194 + 0.598i)25-s + (0.155 + 0.113i)27-s + (−0.256 + 0.789i)29-s + (1.25 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664288 - 0.619632i\)
\(L(\frac12)\) \(\approx\) \(0.664288 - 0.619632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.19 - 2.48i)T \)
good5 \( 1 + (2.30 + 1.67i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.30 + 4.02i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.42 + 1.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.73 + 2.71i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.88 - 5.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.97 + 5.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.54 - 7.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.163 - 0.502i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.527T + 43T^{2} \)
47 \( 1 + (-0.427 - 1.31i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.9 + 7.97i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.73 + 8.42i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.309 - 0.224i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 + (-2.92 - 2.12i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.381 - 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.89 - 5.73i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.28 + 3.83i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (4.92 - 3.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.02179895009478013180507609692, −11.95669889552416230509734138208, −11.25733497321636712340669022603, −10.03140072750779846902239511773, −8.518306992998538216819931188209, −7.62793303750972103654710700846, −6.77704490821969263247705223202, −4.84604201459883330511068146190, −3.87955801023578002104428694474, −1.11046200258126232468007514752, 2.86502907162679550106792156694, 4.28132376253438227824556470886, 5.73415356374875967179216508649, 6.95532648018144515971255713357, 8.538083836680868217347255512317, 9.061603179292636995394485514063, 10.83445418130364867931280641048, 11.41976633361826824401882798301, 12.09290424151202793678758516938, 13.64318321417027490617187705203

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