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Properties

Label 2-132-132.71-c1-0-6
Degree $2$
Conductor $132$
Sign $0.239 - 0.970i$
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  + (1.26 + 0.642i)2-s + (−0.451 + 1.67i)3-s + (1.17 + 1.61i)4-s + (0.587 + 0.190i)5-s + (−1.64 + 1.81i)6-s + (−2.48 − 3.42i)7-s + (0.442 + 2.79i)8-s + (−2.59 − 1.50i)9-s + (0.618 + 0.618i)10-s + (3.30 − 0.224i)11-s + (−3.23 + 1.23i)12-s + (−0.309 − 0.951i)13-s + (−0.937 − 5.91i)14-s + (−0.584 + 0.896i)15-s + (−1.23 + 3.80i)16-s + (5.79 + 1.88i)17-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (−0.260 + 0.965i)3-s + (0.587 + 0.809i)4-s + (0.262 + 0.0854i)5-s + (−0.670 + 0.742i)6-s + (−0.941 − 1.29i)7-s + (0.156 + 0.987i)8-s + (−0.864 − 0.502i)9-s + (0.195 + 0.195i)10-s + (0.997 − 0.0676i)11-s + (−0.934 + 0.356i)12-s + (−0.0857 − 0.263i)13-s + (−0.250 − 1.58i)14-s + (−0.150 + 0.231i)15-s + (−0.309 + 0.951i)16-s + (1.40 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.239 - 0.970i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22923 + 0.963232i\)
\(L(\frac12)\) \(\approx\) \(1.22923 + 0.963232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.642i)T \)
3 \( 1 + (0.451 - 1.67i)T \)
11 \( 1 + (-3.30 + 0.224i)T \)
good5 \( 1 + (-0.587 - 0.190i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.48 + 3.42i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.79 - 1.88i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.08 + 1.5i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + (2.35 + 3.23i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.812 - 0.263i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.66 - 3.38i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.310 - 0.427i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 + (-3.54 - 2.57i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.89 - 1.59i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.73 + 3.44i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.57 + 4.84i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.85iT - 67T^{2} \)
71 \( 1 + (1.57 - 4.84i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.61 - 3.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.92 + 1.60i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.16 - 12.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (3.80 + 11.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.85783102236706673193516650286, −12.54325128958946746750352111381, −11.58417657830012828959397719080, −10.36159876332140626238673706020, −9.626001923357776686853111072807, −7.962324612422268381431786194259, −6.61457954493049863045611096131, −5.72738875654137048527089573487, −4.16043981810941717074981247282, −3.43063443251676265767581064574, 1.90209919313171292284171327879, 3.39499250571953280246355319957, 5.53399332298812593735813647921, 6.08487952684867358157941974829, 7.29466559416740770975059945762, 9.046079380611537839865657902223, 10.03652478746669642973621946024, 11.73004397272576419474880288037, 12.05697109386826020984757549421, 12.85806410711184434791596640501

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