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Properties

Label 2-132-11.5-c1-0-0
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: $\chi_{132} (49, \cdot )$
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

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

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