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Properties

Label 2-66-3.2-c2-0-6
Degree $2$
Conductor $66$
Sign $-0.353 + 0.935i$
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 + (1.06 − 2.80i)3-s − 2.00·4-s − 1.80i·5-s + (−3.96 − 1.49i)6-s + 0.994·7-s + 2.82i·8-s + (−6.75 − 5.95i)9-s − 2.55·10-s + 3.31i·11-s + (−2.12 + 5.61i)12-s + 13.6·13-s − 1.40i·14-s + (−5.07 − 1.91i)15-s + 4.00·16-s − 3.98i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.353 − 0.935i)3-s − 0.500·4-s − 0.361i·5-s + (−0.661 − 0.249i)6-s + 0.142·7-s + 0.353i·8-s + (−0.750 − 0.661i)9-s − 0.255·10-s + 0.301i·11-s + (−0.176 + 0.467i)12-s + 1.04·13-s − 0.100i·14-s + (−0.338 − 0.127i)15-s + 0.250·16-s − 0.234i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $-0.353 + 0.935i$
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: \((2,\ 66,\ (\ :1),\ -0.353 + 0.935i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.727029 - 1.05193i\)
\(L(\frac12)\) \(\approx\) \(0.727029 - 1.05193i\)
\(L(2)\) 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 \( 1 + (-1.06 + 2.80i)T \)
11 \( 1 - 3.31iT \)
good5 \( 1 + 1.80iT - 25T^{2} \)
7 \( 1 - 0.994T + 49T^{2} \)
13 \( 1 - 13.6T + 169T^{2} \)
17 \( 1 + 3.98iT - 289T^{2} \)
19 \( 1 - 21.5T + 361T^{2} \)
23 \( 1 - 41.1iT - 529T^{2} \)
29 \( 1 + 0.886iT - 841T^{2} \)
31 \( 1 + 33.6T + 961T^{2} \)
37 \( 1 - 7.23T + 1.36e3T^{2} \)
41 \( 1 + 24.5iT - 1.68e3T^{2} \)
43 \( 1 + 76.8T + 1.84e3T^{2} \)
47 \( 1 + 44.8iT - 2.20e3T^{2} \)
53 \( 1 - 40.8iT - 2.80e3T^{2} \)
59 \( 1 - 70.4iT - 3.48e3T^{2} \)
61 \( 1 - 79.4T + 3.72e3T^{2} \)
67 \( 1 + 125.T + 4.48e3T^{2} \)
71 \( 1 - 13.7iT - 5.04e3T^{2} \)
73 \( 1 - 32.1T + 5.32e3T^{2} \)
79 \( 1 - 86.1T + 6.24e3T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 + 17.6iT - 7.92e3T^{2} \)
97 \( 1 + 86.1T + 9.40e3T^{2} \)
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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

−13.81184544537626736154298638252, −13.20296300749529042676692860554, −12.06467492176527953475627230980, −11.20339031723857162159145710995, −9.545092484821062089610261006342, −8.507669488379055786980861784916, −7.23338132520444306636999000396, −5.47982332409081330242116717536, −3.40594659011723709174825145958, −1.44096846003937947884255205183, 3.39530798230392836774286559081, 4.94814364184257329693581444701, 6.39889412644787876320912245656, 8.065014741157322473862380100174, 9.015642136881742645550539342213, 10.30271036346195805326970967813, 11.31421686046770409722638296572, 13.10413399377142845619316038279, 14.26653039113327939807440031554, 14.87474297727590927380182262772

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