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Properties

Label 2-132-12.11-c1-0-3
Degree $2$
Conductor $132$
Sign $-0.543 - 0.839i$
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  + (0.835 + 1.14i)2-s + (−1.10 + 1.33i)3-s + (−0.602 + 1.90i)4-s − 0.671i·5-s + (−2.44 − 0.140i)6-s + 2i·7-s + (−2.67 + 0.907i)8-s + (−0.569 − 2.94i)9-s + (0.766 − 0.561i)10-s + 11-s + (−1.88 − 2.90i)12-s + 4.56·13-s + (−2.28 + 1.67i)14-s + (0.897 + 0.740i)15-s + (−3.27 − 2.29i)16-s + 4.40i·17-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)2-s + (−0.636 + 0.771i)3-s + (−0.301 + 0.953i)4-s − 0.300i·5-s + (−0.998 − 0.0574i)6-s + 0.755i·7-s + (−0.947 + 0.320i)8-s + (−0.189 − 0.981i)9-s + (0.242 − 0.177i)10-s + 0.301·11-s + (−0.543 − 0.839i)12-s + 1.26·13-s + (−0.609 + 0.446i)14-s + (0.231 + 0.191i)15-s + (−0.818 − 0.574i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.543 - 0.839i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.543 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552382 + 1.01614i\)
\(L(\frac12)\) \(\approx\) \(0.552382 + 1.01614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.835 - 1.14i)T \)
3 \( 1 + (1.10 - 1.33i)T \)
11 \( 1 - T \)
good5 \( 1 + 0.671iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 4.40iT - 17T^{2} \)
19 \( 1 + 4.56iT - 19T^{2} \)
23 \( 1 - 3.42T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 1.32iT - 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 - 4.93iT - 41T^{2} \)
43 \( 1 + 8.97iT - 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 + 2.78iT - 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 + 3.62T + 61T^{2} \)
67 \( 1 - 0.803iT - 67T^{2} \)
71 \( 1 + 9.70T + 71T^{2} \)
73 \( 1 - 1.47T + 73T^{2} \)
79 \( 1 + 5.37iT - 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−13.65314484815518729855593238040, −12.64888166198612316093924787300, −11.74687529980412671415348050978, −10.73222750621451914148811438554, −9.093242679753065309006389888750, −8.549820025875027462655367514166, −6.70674954707188405974645750663, −5.80097578533273290063758184148, −4.75934059740872893377323416231, −3.47974066924934622290887451391, 1.32094172064530062817605381214, 3.34203064897544385928719067550, 4.92745028176627226655108190589, 6.21255115909336873362516737736, 7.19099907600499454840524862914, 8.829900400820050061609273200919, 10.40261153254389281662844949383, 10.98607539119617025720178456317, 11.94044190863878440654070604092, 12.87404330574674968663624876760

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