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Properties

Label 2-132-11.9-c1-0-1
Degree $2$
Conductor $132$
Sign $0.569 + 0.821i$
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  + (0.809 − 0.587i)3-s + (−1.19 − 3.66i)5-s + (0.190 + 0.138i)7-s + (0.309 − 0.951i)9-s + (3.30 + 0.224i)11-s + (−1.92 + 5.93i)13-s + (−3.11 − 2.26i)15-s + (0.736 + 2.26i)17-s + (4.11 − 2.99i)19-s + 0.236·21-s − 0.236·23-s + (−7.97 + 5.79i)25-s + (−0.309 − 0.951i)27-s + (−3.61 − 2.62i)29-s + (−1.97 + 6.06i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (−0.532 − 1.63i)5-s + (0.0721 + 0.0524i)7-s + (0.103 − 0.317i)9-s + (0.997 + 0.0676i)11-s + (−0.534 + 1.64i)13-s + (−0.805 − 0.584i)15-s + (0.178 + 0.549i)17-s + (0.944 − 0.686i)19-s + 0.0515·21-s − 0.0492·23-s + (−1.59 + 1.15i)25-s + (−0.0594 − 0.183i)27-s + (−0.671 − 0.488i)29-s + (−0.354 + 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03884 - 0.543784i\)
\(L(\frac12)\) \(\approx\) \(1.03884 - 0.543784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.30 - 0.224i)T \)
good5 \( 1 + (1.19 + 3.66i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.190 - 0.138i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.92 - 5.93i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.736 - 2.26i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.11 + 2.99i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + (3.61 + 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.97 - 6.06i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.04 - 2.21i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.66 - 5.56i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.47T + 43T^{2} \)
47 \( 1 + (2.92 - 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.02 + 6.24i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.73 + 1.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.809 + 2.48i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.145T + 67T^{2} \)
71 \( 1 + (0.427 + 1.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.61 + 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.39 + 13.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.78 - 14.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (1.57 - 4.84i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.06763164629855910138162413245, −12.06914631370668899927657244885, −11.57651603177442558553511684328, −9.478502002555345129627145528582, −9.010257560971696671008778223460, −7.937834846671476902925844107696, −6.71467686384439527403880780588, −4.98008397574263834196803367124, −3.92678294200077728056217813896, −1.52861188049291667370098060658, 2.85714431357859294243242972176, 3.80701222166479103886419245729, 5.72463439076555318767189826877, 7.21229331240817175452322730230, 7.84940340176159920000411899183, 9.457829417836284518158699320584, 10.38609749642348924862683196825, 11.24020184708588506911837919634, 12.28112947770593293046945313118, 13.75903316016233931211234401340

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