Properties

Label 2-66-1.1-c1-0-1
Degree $2$
Conductor $66$
Sign $1$
Analytic cond. $0.527012$
Root an. cond. $0.725956$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 2·5-s − 6-s − 4·7-s + 8-s + 9-s + 2·10-s − 11-s − 12-s − 6·13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 2·20-s + 4·21-s − 22-s + 4·23-s − 24-s − 25-s − 6·26-s − 27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.872·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102192530\)
\(L(\frac12)\) \(\approx\) \(1.102192530\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.77427121841588439759624877733, −13.55277453788152412700214007654, −12.76064311034148466780729958842, −11.85600244708104500189303464848, −10.17335307448534281740885152086, −9.626249861423627809345281403043, −7.24765397306950965920878295914, −6.15685276163708486295790955262, −5.03220048416044134767486652990, −2.90847219285821389032814298970, 2.90847219285821389032814298970, 5.03220048416044134767486652990, 6.15685276163708486295790955262, 7.24765397306950965920878295914, 9.626249861423627809345281403043, 10.17335307448534281740885152086, 11.85600244708104500189303464848, 12.76064311034148466780729958842, 13.55277453788152412700214007654, 14.77427121841588439759624877733

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