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Properties

Label 2-66-11.6-c2-0-3
Degree $2$
Conductor $66$
Sign $0.573 + 0.819i$
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 + (1.91 − 5.90i)5-s + (−2.32 − 0.756i)6-s + (1.56 + 2.15i)7-s + (1.66 − 2.28i)8-s + (0.927 + 2.85i)9-s − 8.77i·10-s + (6.10 + 9.15i)11-s − 3.46·12-s + (−7.54 + 2.45i)13-s + (3.04 + 2.21i)14-s + (−8.69 + 6.31i)15-s + (1.23 − 3.80i)16-s + (−7.16 − 2.32i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.467 − 0.339i)3-s + (0.404 − 0.293i)4-s + (0.383 − 1.18i)5-s + (−0.388 − 0.126i)6-s + (0.223 + 0.307i)7-s + (0.207 − 0.286i)8-s + (0.103 + 0.317i)9-s − 0.877i·10-s + (0.554 + 0.831i)11-s − 0.288·12-s + (−0.580 + 0.188i)13-s + (0.217 + 0.158i)14-s + (−0.579 + 0.421i)15-s + (0.0772 − 0.237i)16-s + (−0.421 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(1.79836\)
Root analytic conductor: \(1.34103\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1),\ 0.573 + 0.819i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41179 - 0.735427i\)
\(L(\frac12)\) \(\approx\) \(1.41179 - 0.735427i\)
\(L(2)\) 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 \( 1 + (-6.10 - 9.15i)T \)
good5 \( 1 + (-1.91 + 5.90i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (-1.56 - 2.15i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (7.54 - 2.45i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (7.16 + 2.32i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (21.0 - 28.9i)T + (-111. - 343. i)T^{2} \)
23 \( 1 - 41.2T + 529T^{2} \)
29 \( 1 + (3.90 + 5.37i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (4.73 + 14.5i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (28.9 - 21.0i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-15.8 + 21.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 8.05iT - 1.84e3T^{2} \)
47 \( 1 + (10.8 + 7.89i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (27.8 + 85.5i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (83.3 - 60.5i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (0.171 + 0.0557i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 54.4T + 4.48e3T^{2} \)
71 \( 1 + (16.5 - 51.0i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-69.1 - 95.1i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (110. - 35.7i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (115. + 37.4i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 76.6T + 7.92e3T^{2} \)
97 \( 1 + (49.5 + 152. i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−14.34192619982363601087623262669, −12.93200807230406732630173021263, −12.50590223506450722585440906616, −11.44344001720319539482109660028, −9.955430353288963120476290010629, −8.663983328567759410081782137976, −6.94099242586841815601669519766, −5.50674037657792706139334243790, −4.46483773401976935701876582333, −1.76842235617609702954947919863, 2.99474727827265650930916809709, 4.72512461712018859607257151464, 6.27822357100281492868453360662, 7.12198484373774286585076937540, 9.066663590037125729852071684305, 10.84596764959882182564432104031, 11.05091324555804857688005248315, 12.69161087884839816857176681484, 13.87057506368639524690754735224, 14.75382018283901181775680382200

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