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Properties

Label 2-66-11.2-c2-0-1
Degree $2$
Conductor $66$
Sign $0.970 + 0.242i$
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  + (−1.34 − 0.437i)2-s + (1.40 − 1.01i)3-s + (1.61 + 1.17i)4-s + (1.96 + 6.03i)5-s + (−2.32 + 0.756i)6-s + (5.91 − 8.14i)7-s + (−1.66 − 2.28i)8-s + (0.927 − 2.85i)9-s − 8.97i·10-s + (10.7 + 2.43i)11-s + 3.46·12-s + (−1.07 − 0.348i)13-s + (−11.5 + 8.36i)14-s + (8.89 + 6.46i)15-s + (1.23 + 3.80i)16-s + (−18.8 + 6.12i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.467 − 0.339i)3-s + (0.404 + 0.293i)4-s + (0.392 + 1.20i)5-s + (−0.388 + 0.126i)6-s + (0.845 − 1.16i)7-s + (−0.207 − 0.286i)8-s + (0.103 − 0.317i)9-s − 0.897i·10-s + (0.975 + 0.221i)11-s + 0.288·12-s + (−0.0823 − 0.0267i)13-s + (−0.822 + 0.597i)14-s + (0.592 + 0.430i)15-s + (0.0772 + 0.237i)16-s + (−1.10 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(1.79836\)
Root analytic conductor: \(1.34103\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.14879 - 0.141573i\)
\(L(\frac12)\) \(\approx\) \(1.14879 - 0.141573i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.437i)T \)
3 \( 1 + (-1.40 + 1.01i)T \)
11 \( 1 + (-10.7 - 2.43i)T \)
good5 \( 1 + (-1.96 - 6.03i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-5.91 + 8.14i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (1.07 + 0.348i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (18.8 - 6.12i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-0.867 - 1.19i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 38.3T + 529T^{2} \)
29 \( 1 + (-10.1 + 13.9i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (18.5 - 57.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (19.5 + 14.2i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (19.6 + 27.0i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 42.3iT - 1.84e3T^{2} \)
47 \( 1 + (-48.3 + 35.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (7.40 - 22.7i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (35.0 + 25.4i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (97.0 - 31.5i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 110.T + 4.48e3T^{2} \)
71 \( 1 + (-10.2 - 31.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (26.8 - 37.0i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-73.0 - 23.7i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (67.1 - 21.8i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 53.1T + 7.92e3T^{2} \)
97 \( 1 + (-33.4 + 102. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−14.25090496488735717318911261936, −13.90250710048495061317297652728, −12.13451094252914487430497364624, −10.88167409417136469958407130661, −10.18861443130706097373454820537, −8.714971168291639155664805040718, −7.38782096059832064321662427129, −6.58704801980766935142764380806, −3.87144927826549185668770496130, −1.92003666008824526803182970237, 1.92283121983459928074846119548, 4.64958536952187431831400755164, 6.01836576880122267005215095159, 8.058162895176664508661366539887, 8.922565317074818417080899271097, 9.529991345237817343180824876826, 11.29403921541134676965574140039, 12.27533345895934052670962087355, 13.72620047458768154195157146582, 14.84212141087056779006366904545

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