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Properties

Label 2-132-44.19-c1-0-5
Degree $2$
Conductor $132$
Sign $0.435 - 0.900i$
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  + (0.690 + 1.23i)2-s + (0.951 − 0.309i)3-s + (−1.04 + 1.70i)4-s + (0.725 + 0.527i)5-s + (1.03 + 0.960i)6-s + (0.489 − 1.50i)7-s + (−2.82 − 0.116i)8-s + (0.809 − 0.587i)9-s + (−0.149 + 1.25i)10-s + (−2.62 + 2.03i)11-s + (−0.469 + 1.94i)12-s + (−0.605 − 0.833i)13-s + (2.19 − 0.435i)14-s + (0.853 + 0.277i)15-s + (−1.80 − 3.56i)16-s + (2.49 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.488 + 0.872i)2-s + (0.549 − 0.178i)3-s + (−0.523 + 0.852i)4-s + (0.324 + 0.235i)5-s + (0.423 + 0.392i)6-s + (0.185 − 0.569i)7-s + (−0.999 − 0.0410i)8-s + (0.269 − 0.195i)9-s + (−0.0473 + 0.398i)10-s + (−0.790 + 0.612i)11-s + (−0.135 + 0.561i)12-s + (−0.167 − 0.231i)13-s + (0.587 − 0.116i)14-s + (0.220 + 0.0715i)15-s + (−0.451 − 0.892i)16-s + (0.604 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.435 - 0.900i$
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: \((2,\ 132,\ (\ :1/2),\ 0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28017 + 0.803072i\)
\(L(\frac12)\) \(\approx\) \(1.28017 + 0.803072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.690 - 1.23i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (2.62 - 2.03i)T \)
good5 \( 1 + (-0.725 - 0.527i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.489 + 1.50i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.605 + 0.833i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.49 + 3.42i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.32 + 4.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.34iT - 23T^{2} \)
29 \( 1 + (-0.588 - 0.191i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.11 + 4.29i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.924 - 2.84i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0662 - 0.0215i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.67T + 43T^{2} \)
47 \( 1 + (1.49 - 0.484i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.74 - 7.07i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.81 - 2.53i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.51 - 10.3i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 + (3.76 - 5.18i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.25 + 1.38i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.25 + 4.54i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.5 + 9.08i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.00T + 89T^{2} \)
97 \( 1 + (-11.6 + 8.44i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.59357730211896451445926029973, −12.91533176848310458940176916267, −11.67153781769267513420114046526, −10.13744259024531066977456588482, −9.121071208455566668847344650911, −7.69601149526962873175977968107, −7.23326171418840121738052680123, −5.70295105468239003539569066146, −4.42114655231563760579329402916, −2.83340762685708517579963169085, 2.06301364248299963374280803569, 3.49861115856550553669399885016, 4.99667918448704436773923984589, 6.07986731506645390891310985349, 8.145787734474675961686098296749, 9.032418666057171127263599841360, 10.17187369915468502193843731579, 10.95884152080587545841697478521, 12.36992823073083956025128394223, 12.89533982878654883729026201938

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