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Properties

Label 2-132-33.2-c1-0-0
Degree $2$
Conductor $132$
Sign $-0.822 - 0.568i$
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  + (−1.71 − 0.244i)3-s + (−2.90 + 0.945i)5-s + (−2.14 + 2.95i)7-s + (2.88 + 0.838i)9-s + (−3.28 − 0.421i)11-s + (1.02 + 0.333i)13-s + (5.21 − 0.909i)15-s + (−0.380 − 1.17i)17-s + (−3.04 − 4.19i)19-s + (4.40 − 4.53i)21-s + 6.43i·23-s + (3.52 − 2.56i)25-s + (−4.73 − 2.14i)27-s + (6.00 + 4.36i)29-s + (1.21 − 3.72i)31-s + ⋯
L(s)  = 1  + (−0.989 − 0.141i)3-s + (−1.30 + 0.422i)5-s + (−0.810 + 1.11i)7-s + (0.960 + 0.279i)9-s + (−0.991 − 0.127i)11-s + (0.285 + 0.0926i)13-s + (1.34 − 0.234i)15-s + (−0.0923 − 0.284i)17-s + (−0.698 − 0.961i)19-s + (0.960 − 0.990i)21-s + 1.34i·23-s + (0.704 − 0.512i)25-s + (−0.911 − 0.412i)27-s + (1.11 + 0.809i)29-s + (0.217 − 0.669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0790496 + 0.253263i\)
\(L(\frac12)\) \(\approx\) \(0.0790496 + 0.253263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.244i)T \)
11 \( 1 + (3.28 + 0.421i)T \)
good5 \( 1 + (2.90 - 0.945i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.14 - 2.95i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.02 - 0.333i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.380 + 1.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.04 + 4.19i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.43iT - 23T^{2} \)
29 \( 1 + (-6.00 - 4.36i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.21 + 3.72i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.171 + 0.124i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.20 - 3.77i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 + (-1.87 - 2.57i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (7.80 + 2.53i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.66 + 5.04i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.80 - 1.88i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.24T + 67T^{2} \)
71 \( 1 + (-0.315 + 0.102i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.00 - 11.0i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.26 - 0.410i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.41 - 7.42i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + (-1.96 + 6.04i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.30652111036271327200073451175, −12.52073109208829828462670675671, −11.58397871191036692383310001943, −10.97867438926621327747251923845, −9.674297038729716031339884091212, −8.253131925318020386280000461101, −7.10709125737577554121398336904, −6.03970095382491661361495481383, −4.73325146665534809261800436670, −3.03402654324285425835462740930, 0.30055160008185521298885688408, 3.76503299916163902508728506855, 4.68338448912369804148328434428, 6.30372133838640084013331012860, 7.38107668050095993656213069318, 8.446916345540861166037929430803, 10.33689600683740014300296698013, 10.56319706933951379643984752538, 12.02885734999191789415964630477, 12.58316384044988383624477564793

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