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Properties

Label 2-132-44.7-c1-0-11
Degree $2$
Conductor $132$
Sign $-0.0233 + 0.999i$
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  + (0.960 − 1.03i)2-s + (−0.951 − 0.309i)3-s + (−0.154 − 1.99i)4-s + (0.725 − 0.527i)5-s + (−1.23 + 0.690i)6-s + (−0.489 − 1.50i)7-s + (−2.21 − 1.75i)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 + (−2.03 − 0.939i)14-s + (−0.853 + 0.277i)15-s + (−3.95 + 0.616i)16-s + (2.49 + 3.42i)17-s + ⋯
L(s)  = 1  + (0.679 − 0.733i)2-s + (−0.549 − 0.178i)3-s + (−0.0772 − 0.997i)4-s + (0.324 − 0.235i)5-s + (−0.503 + 0.281i)6-s + (−0.185 − 0.569i)7-s + (−0.784 − 0.620i)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 + (−0.543 − 0.251i)14-s + (−0.220 + 0.0715i)15-s + (−0.988 + 0.154i)16-s + (0.604 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0233 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0233 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.0233 + 0.999i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.0233 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914861 - 0.936524i\)
\(L(\frac12)\) \(\approx\) \(0.914861 - 0.936524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.960 + 1.03i)T \)
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

−13.03265933066676190615241046821, −11.97130183540152827110532006823, −11.24690499312175512863775172195, −10.03991899182248649301991961481, −9.319362644020970081183568441976, −7.31616323012965950971366220675, −6.14937728143696833413889843989, −4.97468268610159991734809311538, −3.68591739356461522961272995918, −1.54982171490119697601302960552, 3.07663944947565345156821982212, 4.67213843164692992893482417079, 5.90850261052014054554968038858, 6.60067707836188088704115385615, 8.073833419302570816180861827597, 9.246624737799573756912618045561, 10.54020737753688196513263726454, 11.98746894697092861839897008160, 12.32141356175655378441450845139, 13.81889482496847319623492963528

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