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Properties

Label 2-66-3.2-c2-0-3
Degree $2$
Conductor $66$
Sign $0.760 - 0.649i$
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 + (2.28 − 1.94i)3-s − 2.00·4-s + 5.56i·5-s + (2.75 + 3.22i)6-s + 9.94·7-s − 2.82i·8-s + (1.40 − 8.88i)9-s − 7.87·10-s + 3.31i·11-s + (−4.56 + 3.89i)12-s − 18.1·13-s + 14.0i·14-s + (10.8 + 12.6i)15-s + 4.00·16-s − 18.4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.760 − 0.649i)3-s − 0.500·4-s + 1.11i·5-s + (0.459 + 0.537i)6-s + 1.42·7-s − 0.353i·8-s + (0.156 − 0.987i)9-s − 0.787·10-s + 0.301i·11-s + (−0.380 + 0.324i)12-s − 1.39·13-s + 1.00i·14-s + (0.722 + 0.846i)15-s + 0.250·16-s − 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.760 - 0.649i$
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: \((2,\ 66,\ (\ :1),\ 0.760 - 0.649i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40472 + 0.518179i\)
\(L(\frac12)\) \(\approx\) \(1.40472 + 0.518179i\)
\(L(2)\) 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 \( 1 + (-2.28 + 1.94i)T \)
11 \( 1 - 3.31iT \)
good5 \( 1 - 5.56iT - 25T^{2} \)
7 \( 1 - 9.94T + 49T^{2} \)
13 \( 1 + 18.1T + 169T^{2} \)
17 \( 1 + 18.4iT - 289T^{2} \)
19 \( 1 + 29.8T + 361T^{2} \)
23 \( 1 + 13.1iT - 529T^{2} \)
29 \( 1 - 15.8iT - 841T^{2} \)
31 \( 1 - 22.6T + 961T^{2} \)
37 \( 1 - 21.8T + 1.36e3T^{2} \)
41 \( 1 - 9.61iT - 1.68e3T^{2} \)
43 \( 1 + 44.2T + 1.84e3T^{2} \)
47 \( 1 - 52.9iT - 2.20e3T^{2} \)
53 \( 1 + 51.1iT - 2.80e3T^{2} \)
59 \( 1 - 54.9iT - 3.48e3T^{2} \)
61 \( 1 - 24.0T + 3.72e3T^{2} \)
67 \( 1 - 107.T + 4.48e3T^{2} \)
71 \( 1 - 46.5iT - 5.04e3T^{2} \)
73 \( 1 - 6.88T + 5.32e3T^{2} \)
79 \( 1 + 63.7T + 6.24e3T^{2} \)
83 \( 1 - 57.4iT - 6.88e3T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + 87.7T + 9.40e3T^{2} \)
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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

−14.55584939589453271412125712933, −14.22832197442532401120641282205, −12.73919869925378244592120137542, −11.45569069889441956704078948182, −10.00585875375875894167364790566, −8.519550802367891379618802273973, −7.50497437466398044786515643292, −6.68119788233287057589348902460, −4.67936666833230440649765139993, −2.47236037408845179453311840154, 2.01030323929040535059957533044, 4.24758823722394731448346540587, 5.08965766727178047427208545900, 8.050463990223636041418106458534, 8.640724289265122667742573638031, 9.908390201338065823182834369824, 11.02774427476990364359669872210, 12.27523717938173084545821361465, 13.33271135097267491806136974762, 14.52754332477918949460758077277

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