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Properties

Label 2-132-44.19-c1-0-4
Degree $2$
Conductor $132$
Sign $0.715 - 0.698i$
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  + (−0.877 + 1.10i)2-s + (0.951 − 0.309i)3-s + (−0.461 − 1.94i)4-s + (1.42 + 1.03i)5-s + (−0.491 + 1.32i)6-s + (0.160 − 0.492i)7-s + (2.56 + 1.19i)8-s + (0.809 − 0.587i)9-s + (−2.39 + 0.672i)10-s + (2.95 + 1.50i)11-s + (−1.04 − 1.70i)12-s + (−0.761 − 1.04i)13-s + (0.406 + 0.609i)14-s + (1.67 + 0.543i)15-s + (−3.57 + 1.79i)16-s + (−4.11 + 5.66i)17-s + ⋯
L(s)  = 1  + (−0.620 + 0.784i)2-s + (0.549 − 0.178i)3-s + (−0.230 − 0.973i)4-s + (0.636 + 0.462i)5-s + (−0.200 + 0.541i)6-s + (0.0605 − 0.186i)7-s + (0.906 + 0.422i)8-s + (0.269 − 0.195i)9-s + (−0.757 + 0.212i)10-s + (0.890 + 0.454i)11-s + (−0.300 − 0.493i)12-s + (−0.211 − 0.290i)13-s + (0.108 + 0.162i)14-s + (0.432 + 0.140i)15-s + (−0.893 + 0.448i)16-s + (−0.998 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.715 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951705 + 0.387344i\)
\(L(\frac12)\) \(\approx\) \(0.951705 + 0.387344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.877 - 1.10i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-2.95 - 1.50i)T \)
good5 \( 1 + (-1.42 - 1.03i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.160 + 0.492i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.761 + 1.04i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.11 - 5.66i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.159 + 0.490i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.57iT - 23T^{2} \)
29 \( 1 + (5.04 + 1.64i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.919 - 1.26i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.66 + 8.19i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.547 - 0.177i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.40T + 43T^{2} \)
47 \( 1 + (9.60 - 3.12i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.50 - 4.00i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.70 - 0.555i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.49 - 2.05i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.51iT - 67T^{2} \)
71 \( 1 + (3.77 - 5.19i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.83 - 3.19i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.1 + 7.40i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.33 + 3.14i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (11.2 - 8.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.70290076063056488815195058801, −12.66800479004976194040835282760, −10.96467407757681909726143007459, −10.06870640275433043196168423816, −9.104527235506140365554767645561, −8.130016874475492692585909503838, −6.87048360186399384038061315019, −6.12231060610216080520292553842, −4.32634744885059601433938439829, −2.01178126143028611473409328442, 1.80515486719534914218713300381, 3.42644872969528068339931241858, 4.95498455698199562384537438933, 6.86765424113239528769251876085, 8.232500782231216843331696857471, 9.369527658868964330715136369087, 9.564760480950752513366266098365, 11.19296025818836334055767582231, 11.88902942558278056329880454534, 13.34500516327846460443560558804

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