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Properties

Label 2-66-33.2-c1-0-3
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: $\chi_{66} (35, \cdot )$
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

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

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