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Properties

Label 2-66-33.17-c1-0-0
Degree $2$
Conductor $66$
Sign $0.0744 - 0.997i$
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  + (0.309 + 0.951i)2-s + (−1.20 + 1.24i)3-s + (−0.809 + 0.587i)4-s + (0.897 + 0.291i)5-s + (−1.55 − 0.763i)6-s + (0.897 + 1.23i)7-s + (−0.809 − 0.587i)8-s + (−0.0885 − 2.99i)9-s + 0.943i·10-s + (−0.151 − 3.31i)11-s + (0.245 − 1.71i)12-s + (4.03 − 1.30i)13-s + (−0.897 + 1.23i)14-s + (−1.44 + 0.763i)15-s + (0.309 − 0.951i)16-s + (0.906 − 2.78i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.696 + 0.717i)3-s + (−0.404 + 0.293i)4-s + (0.401 + 0.130i)5-s + (−0.634 − 0.311i)6-s + (0.339 + 0.466i)7-s + (−0.286 − 0.207i)8-s + (−0.0295 − 0.999i)9-s + 0.298i·10-s + (−0.0457 − 0.998i)11-s + (0.0709 − 0.494i)12-s + (1.11 − 0.363i)13-s + (−0.239 + 0.330i)14-s + (−0.373 + 0.197i)15-s + (0.0772 − 0.237i)16-s + (0.219 − 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0744 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0744 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.0744 - 0.997i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.0744 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629055 + 0.583828i\)
\(L(\frac12)\) \(\approx\) \(0.629055 + 0.583828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (1.20 - 1.24i)T \)
11 \( 1 + (0.151 + 3.31i)T \)
good5 \( 1 + (-0.897 - 0.291i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.897 - 1.23i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.03 + 1.30i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.906 + 2.78i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.16 - 5.73i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.18iT - 23T^{2} \)
29 \( 1 + (5.17 - 3.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.68 + 8.27i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.28 - 1.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.92 - 4.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.34iT - 43T^{2} \)
47 \( 1 + (-5.58 + 7.68i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.62 - 1.50i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.74 + 3.77i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.685 + 0.222i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.89T + 67T^{2} \)
71 \( 1 + (9.47 + 3.07i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.03 - 1.41i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.56 - 0.508i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.90 - 5.85i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (-3.54 - 10.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−15.23159172321759427002782449567, −14.27157842105267393531518969152, −13.06563984449599405010938932433, −11.69879433629594096402868699076, −10.67636577981226015967174000416, −9.337076836912607065278745238114, −8.137805307345306152817497446186, −6.17332420634124999948502858908, −5.54323303195961987976600529875, −3.78607930167710716546258184004, 1.79085220651777291868484564655, 4.40559468603441964692091211678, 5.90068899363987935576815801707, 7.28759690884763704045301047487, 8.917167038965020160171224772862, 10.50094266058977174620367467844, 11.21881473865935706768216890802, 12.50537561084297759625712105727, 13.22948029403409654335198307326, 14.20690022980768096565397197592

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