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Properties

Label 2-132-12.11-c1-0-15
Degree $2$
Conductor $132$
Sign $-0.577 + 0.816i$
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  − 1.41i·2-s + (−1 + 1.41i)3-s − 2.00·4-s − 2.82i·5-s + (2.00 + 1.41i)6-s − 4.24i·7-s + 2.82i·8-s + (−1.00 − 2.82i)9-s − 4.00·10-s − 11-s + (2.00 − 2.82i)12-s + 2·13-s − 6·14-s + (4.00 + 2.82i)15-s + 4.00·16-s + 1.41i·17-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.577 + 0.816i)3-s − 1.00·4-s − 1.26i·5-s + (0.816 + 0.577i)6-s − 1.60i·7-s + 1.00i·8-s + (−0.333 − 0.942i)9-s − 1.26·10-s − 0.301·11-s + (0.577 − 0.816i)12-s + 0.554·13-s − 1.60·14-s + (1.03 + 0.730i)15-s + 1.00·16-s + 0.342i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348582 - 0.673409i\)
\(L(\frac12)\) \(\approx\) \(0.348582 - 0.673409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1 - 1.41i)T \)
11 \( 1 + T \)
good5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−12.84962431339103424117195072052, −11.78575737564025760865233305256, −10.76964692868348381413460559278, −10.08109787471044109758193579791, −9.084669387968693042230345945222, −7.911563722714694533301076064802, −5.78102983663802657244753705477, −4.49553537580106863121085449071, −3.84020730261094744531162129257, −0.920927979099957066703298060197, 2.75912214925917698388358164521, 5.16416956866915692204073770275, 6.16415078936256485591237541466, 6.90914133448345252975866874203, 8.069472940478806801029120282064, 9.165220232686028115997616358810, 10.66814135977600951713538780105, 11.70820285436315356789457690899, 12.75569804133578967129180667184, 13.73427918347347374462621800659

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