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Properties

Label 2-66-33.26-c2-0-1
Degree $2$
Conductor $66$
Sign $0.518 - 0.855i$
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  + (−0.831 + 1.14i)2-s + (2.91 + 0.712i)3-s + (−0.618 − 1.90i)4-s + (2.69 + 3.71i)5-s + (−3.23 + 2.74i)6-s + (−1.21 − 3.74i)7-s + (2.68 + 0.874i)8-s + (7.98 + 4.15i)9-s − 6.49·10-s + (−6.59 + 8.80i)11-s + (−0.444 − 5.98i)12-s + (2.32 + 1.69i)13-s + (5.30 + 1.72i)14-s + (5.21 + 12.7i)15-s + (−3.23 + 2.35i)16-s + (−12.4 − 17.1i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.971 + 0.237i)3-s + (−0.154 − 0.475i)4-s + (0.539 + 0.742i)5-s + (−0.539 + 0.456i)6-s + (−0.174 − 0.535i)7-s + (0.336 + 0.109i)8-s + (0.887 + 0.461i)9-s − 0.649·10-s + (−0.599 + 0.800i)11-s + (−0.0370 − 0.498i)12-s + (0.179 + 0.130i)13-s + (0.378 + 0.123i)14-s + (0.347 + 0.849i)15-s + (−0.202 + 0.146i)16-s + (−0.732 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(1.79836\)
Root analytic conductor: \(1.34103\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1),\ 0.518 - 0.855i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17078 + 0.659644i\)
\(L(\frac12)\) \(\approx\) \(1.17078 + 0.659644i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 1.14i)T \)
3 \( 1 + (-2.91 - 0.712i)T \)
11 \( 1 + (6.59 - 8.80i)T \)
good5 \( 1 + (-2.69 - 3.71i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (1.21 + 3.74i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-2.32 - 1.69i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (12.4 + 17.1i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-7.91 + 24.3i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 20.3iT - 529T^{2} \)
29 \( 1 + (23.0 - 7.48i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (31.0 + 22.5i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-20.6 - 63.4i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-36.7 - 11.9i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 32.3T + 1.84e3T^{2} \)
47 \( 1 + (-5.00 - 1.62i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (31.3 - 43.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (37.3 - 12.1i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (66.2 - 48.1i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 113.T + 4.48e3T^{2} \)
71 \( 1 + (62.5 + 86.0i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-30.2 - 92.9i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-42.6 - 31.0i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (30.0 + 41.4i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 0.204iT - 7.92e3T^{2} \)
97 \( 1 + (5.42 + 3.94i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−14.87698376792846342059341761661, −13.85471373454298732599513132275, −13.13031837200768786304684261167, −10.96719806916651193415001701787, −9.949373960743889307872763482413, −9.094271253772699468543585051295, −7.57603178016339417453226667687, −6.73314298577353675335521182248, −4.67261569582853492277838288236, −2.57936230331323672871333752847, 1.82781217901650830188245500948, 3.58200430212945380427741962600, 5.72758423720680632772442596118, 7.76726316067065051646660990735, 8.787042396021474927469923649363, 9.544980009445735697224257060564, 10.90946926907622731557096927015, 12.55720362342444918948028863683, 13.08655633734071990518455890301, 14.18416598786219149061301264647

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