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Properties

Label 2-66-3.2-c2-0-1
Degree $2$
Conductor $66$
Sign $0.520 - 0.854i$
Analytic cond. $1.79836$
Root an. cond. $1.34103$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−1.56 + 2.56i)3-s − 2.00·4-s + 8.92i·5-s + (3.62 + 2.20i)6-s + 5.69·7-s + 2.82i·8-s + (−4.12 − 7.99i)9-s + 12.6·10-s + 3.31i·11-s + (3.12 − 5.12i)12-s − 6.25·13-s − 8.05i·14-s + (−22.8 − 13.9i)15-s + 4.00·16-s + 10.0i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.520 + 0.854i)3-s − 0.500·4-s + 1.78i·5-s + (0.603 + 0.367i)6-s + 0.813·7-s + 0.353i·8-s + (−0.458 − 0.888i)9-s + 1.26·10-s + 0.301i·11-s + (0.260 − 0.427i)12-s − 0.480·13-s − 0.575i·14-s + (−1.52 − 0.928i)15-s + 0.250·16-s + 0.592i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.520 - 0.854i$
Analytic conductor: \(1.79836\)
Root analytic conductor: \(1.34103\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1),\ 0.520 - 0.854i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.867797 + 0.487558i\)
\(L(\frac12)\) \(\approx\) \(0.867797 + 0.487558i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1.56 - 2.56i)T \)
11 \( 1 - 3.31iT \)
good5 \( 1 - 8.92iT - 25T^{2} \)
7 \( 1 - 5.69T + 49T^{2} \)
13 \( 1 + 6.25T + 169T^{2} \)
17 \( 1 - 10.0iT - 289T^{2} \)
19 \( 1 - 27.3T + 361T^{2} \)
23 \( 1 + 20.7iT - 529T^{2} \)
29 \( 1 + 36.4iT - 841T^{2} \)
31 \( 1 - 21.8T + 961T^{2} \)
37 \( 1 - 31.2T + 1.36e3T^{2} \)
41 \( 1 - 10.9iT - 1.68e3T^{2} \)
43 \( 1 - 60.9T + 1.84e3T^{2} \)
47 \( 1 - 6.25iT - 2.20e3T^{2} \)
53 \( 1 - 97.0iT - 2.80e3T^{2} \)
59 \( 1 + 26.2iT - 3.48e3T^{2} \)
61 \( 1 + 73.4T + 3.72e3T^{2} \)
67 \( 1 + 29.9T + 4.48e3T^{2} \)
71 \( 1 + 69.6iT - 5.04e3T^{2} \)
73 \( 1 - 93.9T + 5.32e3T^{2} \)
79 \( 1 + 119.T + 6.24e3T^{2} \)
83 \( 1 + 7.11iT - 6.88e3T^{2} \)
89 \( 1 + 137. iT - 7.92e3T^{2} \)
97 \( 1 - 53.1T + 9.40e3T^{2} \)
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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

−14.71042540376685976587653146304, −14.00052469862738528498624463338, −12.06079233118900700768363181128, −11.22225706389236854850983296090, −10.47438463406109894456221848304, −9.590303039449382402282922707121, −7.68554325098975723427654117590, −6.07218122274454711226522626150, −4.39203202945119923520182628609, −2.83809998563463766425233171883, 1.09936347564716581461389610745, 4.86901142045674512078820341232, 5.55295376239825156664501150001, 7.40508207199345505641257891693, 8.286461198103851910929779299407, 9.433895327130940630927355493092, 11.47677710803095950402549889276, 12.33747719567815152661391604718, 13.34460318278516758492240478937, 14.18890788412120767387142990702

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