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Properties

Label 2-132-44.35-c1-0-0
Degree $2$
Conductor $132$
Sign $0.371 - 0.928i$
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  + (−1.41 − 0.00375i)2-s + (0.587 + 0.809i)3-s + (1.99 + 0.0106i)4-s + (0.907 + 2.79i)5-s + (−0.828 − 1.14i)6-s + (−2.11 − 1.53i)7-s + (−2.82 − 0.0225i)8-s + (−0.309 + 0.951i)9-s + (−1.27 − 3.95i)10-s + (0.900 + 3.19i)11-s + (1.16 + 1.62i)12-s + (4.58 + 1.48i)13-s + (2.98 + 2.18i)14-s + (−1.72 + 2.37i)15-s + (3.99 + 0.0424i)16-s + (−1.58 + 0.515i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00265i)2-s + (0.339 + 0.467i)3-s + (0.999 + 0.00530i)4-s + (0.405 + 1.24i)5-s + (−0.338 − 0.467i)6-s + (−0.800 − 0.581i)7-s + (−0.999 − 0.00796i)8-s + (−0.103 + 0.317i)9-s + (−0.402 − 1.25i)10-s + (0.271 + 0.962i)11-s + (0.336 + 0.468i)12-s + (1.27 + 0.413i)13-s + (0.799 + 0.583i)14-s + (−0.445 + 0.613i)15-s + (0.999 + 0.0106i)16-s + (−0.384 + 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.662793 + 0.448636i\)
\(L(\frac12)\) \(\approx\) \(0.662793 + 0.448636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.00375i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.900 - 3.19i)T \)
good5 \( 1 + (-0.907 - 2.79i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.11 + 1.53i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.58 - 1.48i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.58 - 0.515i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.68 - 1.22i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.78iT - 23T^{2} \)
29 \( 1 + (-3.53 + 4.86i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.86 - 1.57i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.62 + 5.54i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.990 + 1.36i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.84T + 43T^{2} \)
47 \( 1 + (2.67 + 3.67i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.23 + 3.79i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.88 - 5.34i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.319 - 0.103i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.0794iT - 67T^{2} \)
71 \( 1 + (-12.0 + 3.91i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.24 + 5.84i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.97 - 9.14i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.35 - 4.16i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.42T + 89T^{2} \)
97 \( 1 + (4.48 - 13.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.68733004176761138382477148114, −12.26021223733685571611690954064, −10.78969701894264232513237256539, −10.40755021753993225861854013006, −9.482904067391339217157968816423, −8.347905238072024396845777044452, −6.84488156222696747315556652508, −6.40122354983925969625298686906, −3.84570715242934981382085565832, −2.40720992602539844023344049274, 1.25607118652587686070830206193, 3.18329847202590718482204167675, 5.65659277652307935666377645694, 6.55173019590390103608361024851, 8.212644242288945524487048769905, 8.838791944224306001847240021810, 9.532811789018782251759361878879, 10.98207272672017698688406843269, 12.09570580189250290114165296437, 13.01109040741288126706083928565

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