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Properties

Label 2-132-12.11-c1-0-9
Degree $2$
Conductor $132$
Sign $0.557 - 0.830i$
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  + (0.621 + 1.27i)2-s + (1.72 − 0.121i)3-s + (−1.22 + 1.57i)4-s − 2.24i·5-s + (1.22 + 2.11i)6-s + 2i·7-s + (−2.76 − 0.578i)8-s + (2.97 − 0.419i)9-s + (2.84 − 1.39i)10-s − 11-s + (−1.92 + 2.87i)12-s − 5.08·13-s + (−2.54 + 1.24i)14-s + (−0.272 − 3.87i)15-s + (−0.985 − 3.87i)16-s − 6.91i·17-s + ⋯
L(s)  = 1  + (0.439 + 0.898i)2-s + (0.997 − 0.0700i)3-s + (−0.613 + 0.789i)4-s − 1.00i·5-s + (0.501 + 0.865i)6-s + 0.755i·7-s + (−0.978 − 0.204i)8-s + (0.990 − 0.139i)9-s + (0.900 − 0.440i)10-s − 0.301·11-s + (−0.557 + 0.830i)12-s − 1.40·13-s + (−0.679 + 0.332i)14-s + (−0.0702 − 1.00i)15-s + (−0.246 − 0.969i)16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.557 - 0.830i$
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: \((2,\ 132,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39052 + 0.741634i\)
\(L(\frac12)\) \(\approx\) \(1.39052 + 0.741634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.621 - 1.27i)T \)
3 \( 1 + (-1.72 + 0.121i)T \)
11 \( 1 + T \)
good5 \( 1 + 2.24iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + 6.91iT - 17T^{2} \)
19 \( 1 - 5.08iT - 19T^{2} \)
23 \( 1 + 0.859T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 0.222T + 37T^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 + 1.82iT - 43T^{2} \)
47 \( 1 - 1.40T + 47T^{2} \)
53 \( 1 - 6.59iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 14.5iT - 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 + 2.67iT - 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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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

−13.61660382227941015539907706486, −12.53602904550215298747555273777, −12.10487374279603069606807857534, −9.714417900891332785184432343430, −9.057426599020833903038626562503, −8.056224676919056678547149866411, −7.18057543262669329154372411844, −5.45703656168350639058203116134, −4.50108797081744048398302667043, −2.75503106367337222429388167405, 2.28433216965091170069992766204, 3.45351851202256785964265269600, 4.68010223428343197462861938352, 6.63460365997754383478178637469, 7.80951835859178636456759100923, 9.232373611339954480854245203440, 10.31805487911516279499340465532, 10.74939659939183536183609278910, 12.26986037018122951768404802663, 13.22488477450052438198521483133

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