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Properties

Label 2-132-44.39-c1-0-2
Degree $2$
Conductor $132$
Sign $0.0184 - 0.999i$
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  + (0.544 + 1.30i)2-s + (0.587 − 0.809i)3-s + (−1.40 + 1.42i)4-s + (−1.21 + 3.73i)5-s + (1.37 + 0.326i)6-s + (1.21 − 0.882i)7-s + (−2.62 − 1.05i)8-s + (−0.309 − 0.951i)9-s + (−5.54 + 0.452i)10-s + (3.29 − 0.367i)11-s + (0.324 + 1.97i)12-s + (3.86 − 1.25i)13-s + (1.81 + 1.10i)14-s + (2.31 + 3.18i)15-s + (−0.0457 − 3.99i)16-s + (−2.39 − 0.778i)17-s + ⋯
L(s)  = 1  + (0.385 + 0.922i)2-s + (0.339 − 0.467i)3-s + (−0.703 + 0.711i)4-s + (−0.543 + 1.67i)5-s + (0.561 + 0.133i)6-s + (0.458 − 0.333i)7-s + (−0.927 − 0.374i)8-s + (−0.103 − 0.317i)9-s + (−1.75 + 0.142i)10-s + (0.993 − 0.110i)11-s + (0.0935 + 0.569i)12-s + (1.07 − 0.348i)13-s + (0.484 + 0.295i)14-s + (0.596 + 0.821i)15-s + (−0.0114 − 0.999i)16-s + (−0.580 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0184 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0184 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.0184 - 0.999i$
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: \((2,\ 132,\ (\ :1/2),\ 0.0184 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923549 + 0.906703i\)
\(L(\frac12)\) \(\approx\) \(0.923549 + 0.906703i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.544 - 1.30i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-3.29 + 0.367i)T \)
good5 \( 1 + (1.21 - 3.73i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.21 + 0.882i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.86 + 1.25i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.39 + 0.778i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.06 + 2.95i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.73iT - 23T^{2} \)
29 \( 1 + (-4.75 - 6.53i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.284 + 0.0926i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.122 - 0.0888i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.79 + 2.46i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.820T + 43T^{2} \)
47 \( 1 + (5.03 - 6.93i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.56 + 4.82i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.75 + 5.16i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-6.42 - 2.08i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.59iT - 67T^{2} \)
71 \( 1 + (14.4 + 4.70i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.670 + 0.922i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.475 - 1.46i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.63 - 5.02i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (-0.674 - 2.07i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.96763138053385840133288543960, −12.78825734352911963863068732867, −11.49485526844885076095392124311, −10.67166881441859519653943661236, −8.906425513445931131991603226547, −7.953873420915052244535582961570, −6.81002644645463090681479440588, −6.39508031698802247493192595239, −4.24900834114015531357448352732, −3.07070786274359628490737418805, 1.57723603285282836412090425069, 3.92829026168389714409134524330, 4.54765142537146461128520739684, 5.91007474596986707687386226356, 8.426414389106137977993286395727, 8.769054214068891019966436689991, 9.853371022903705257209765331017, 11.32945463334183817260933542400, 11.92073570691167539198781935124, 12.94453209022318900303450529239

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