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Properties

Label 2-66-33.32-c1-0-3
Degree $2$
Conductor $66$
Sign $0.870 + 0.492i$
Analytic cond. $0.527012$
Root an. cond. $0.725956$
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  + 2-s + (−1 − 1.41i)3-s + 4-s − 1.41i·5-s + (−1 − 1.41i)6-s + 4.24i·7-s + 8-s + (−1.00 + 2.82i)9-s − 1.41i·10-s + (−3 + 1.41i)11-s + (−1 − 1.41i)12-s − 4.24i·13-s + 4.24i·14-s + (−2.00 + 1.41i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.577 − 0.816i)3-s + 0.5·4-s − 0.632i·5-s + (−0.408 − 0.577i)6-s + 1.60i·7-s + 0.353·8-s + (−0.333 + 0.942i)9-s − 0.447i·10-s + (−0.904 + 0.426i)11-s + (−0.288 − 0.408i)12-s − 1.17i·13-s + 1.13i·14-s + (−0.516 + 0.365i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.870 + 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04468 - 0.275004i\)
\(L(\frac12)\) \(\approx\) \(1.04468 - 0.275004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1 + 1.41i)T \)
11 \( 1 + (3 - 1.41i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 9.89iT - 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 4.24iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
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

−14.85710888317907550720454329961, −13.18973373596080614623080226875, −12.65757993538780706362460002277, −11.90639861748852524710485921961, −10.60185551644306953262558480872, −8.744716998185868300578536231823, −7.51152034717309059683196971856, −5.78406228668279500121033588218, −5.19087913253442988808052572426, −2.44891858941524232192281638879, 3.52680226914098668103398441491, 4.71396347176633781118207246611, 6.31696351379615808866233753130, 7.48189790337376074635050464047, 9.628066559626933826721607268286, 10.86966043714630934495953309802, 11.18580052828237042573334328733, 12.89239506528140114382587311790, 14.02870212106344477580131069379, 14.78068551447631157172825048546

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