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Properties

Label 2-132-44.39-c1-0-5
Degree $2$
Conductor $132$
Sign $0.247 + 0.968i$
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  + (−1.02 − 0.973i)2-s + (0.587 − 0.809i)3-s + (0.105 + 1.99i)4-s + (−0.210 + 0.646i)5-s + (−1.39 + 0.257i)6-s + (4.14 − 3.01i)7-s + (1.83 − 2.15i)8-s + (−0.309 − 0.951i)9-s + (0.845 − 0.459i)10-s + (−2.98 − 1.45i)11-s + (1.67 + 1.08i)12-s + (−2.14 + 0.697i)13-s + (−7.18 − 0.944i)14-s + (0.399 + 0.550i)15-s + (−3.97 + 0.420i)16-s + (3.12 + 1.01i)17-s + ⋯
L(s)  = 1  + (−0.725 − 0.688i)2-s + (0.339 − 0.467i)3-s + (0.0526 + 0.998i)4-s + (−0.0939 + 0.289i)5-s + (−0.567 + 0.105i)6-s + (1.56 − 1.13i)7-s + (0.649 − 0.760i)8-s + (−0.103 − 0.317i)9-s + (0.267 − 0.145i)10-s + (−0.899 − 0.437i)11-s + (0.484 + 0.314i)12-s + (−0.595 + 0.193i)13-s + (−1.91 − 0.252i)14-s + (0.103 + 0.142i)15-s + (−0.994 + 0.105i)16-s + (0.758 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721266 - 0.560097i\)
\(L(\frac12)\) \(\approx\) \(0.721266 - 0.560097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 + 0.973i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (2.98 + 1.45i)T \)
good5 \( 1 + (0.210 - 0.646i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-4.14 + 3.01i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.14 - 0.697i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.12 - 1.01i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.65 - 1.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.00iT - 23T^{2} \)
29 \( 1 + (-3.13 - 4.31i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.70 - 2.17i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.02 - 5.10i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.57 - 2.17i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.07T + 43T^{2} \)
47 \( 1 + (5.92 - 8.15i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.123 - 0.379i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.22 - 5.80i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.25 + 2.35i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.95iT - 67T^{2} \)
71 \( 1 + (2.41 + 0.783i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.107 - 0.147i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.58 + 7.94i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.66 + 11.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.32T + 89T^{2} \)
97 \( 1 + (0.580 + 1.78i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.94898160642080522553537115929, −11.85998422573033331992894564147, −10.86879271605753124960171445989, −10.26024782929947431816052550647, −8.682241933745348229703059011856, −7.78601537128999373290264794160, −7.17401403974348075125182268887, −4.85720958158193692300622207075, −3.23562312714936465153615292084, −1.47364881998944011597609013764, 2.19791324731992159315229141492, 4.92718864076864015577839047730, 5.43415225342763685523570847420, 7.47512377297575393257422964810, 8.218136650062384171021153109670, 9.094708740646512646881157423170, 10.17425636347850045714857737326, 11.24690274258813315555913130188, 12.32220446456922621016243939453, 13.95038092848907859959168162562

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