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Properties

Label 2-66-33.8-c1-0-1
Degree $2$
Conductor $66$
Sign $0.999 + 0.00662i$
Analytic cond. $0.527012$
Root an. cond. $0.725956$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 + 1.70i)3-s + (0.309 − 0.951i)4-s + (0.442 − 0.609i)5-s + (0.751 + 1.56i)6-s + (−0.442 − 0.143i)7-s + (−0.309 − 0.951i)8-s + (−2.80 − 1.05i)9-s − 0.753i·10-s + (−2.46 + 2.21i)11-s + (1.52 + 0.820i)12-s + (−3.12 − 4.29i)13-s + (−0.442 + 0.143i)14-s + (0.901 + 0.942i)15-s + (−0.809 − 0.587i)16-s + (2.99 + 2.17i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.178 + 0.983i)3-s + (0.154 − 0.475i)4-s + (0.197 − 0.272i)5-s + (0.306 + 0.637i)6-s + (−0.167 − 0.0543i)7-s + (−0.109 − 0.336i)8-s + (−0.936 − 0.351i)9-s − 0.238i·10-s + (−0.744 + 0.668i)11-s + (0.440 + 0.236i)12-s + (−0.865 − 1.19i)13-s + (−0.118 + 0.0384i)14-s + (0.232 + 0.243i)15-s + (−0.202 − 0.146i)16-s + (0.726 + 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.999 + 0.00662i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.999 + 0.00662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08673 - 0.00360147i\)
\(L(\frac12)\) \(\approx\) \(1.08673 - 0.00360147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 1.70i)T \)
11 \( 1 + (2.46 - 2.21i)T \)
good5 \( 1 + (-0.442 + 0.609i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.442 + 0.143i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.12 + 4.29i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.99 - 2.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.02 + 0.659i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.24iT - 23T^{2} \)
29 \( 1 + (-3.09 + 9.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.96 + 2.15i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.16 - 6.66i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.0135 - 0.0416i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.49iT - 43T^{2} \)
47 \( 1 + (-3.03 + 0.987i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.00 - 4.13i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (11.0 + 3.59i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.31 + 3.19i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + (-0.527 + 0.726i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.32 - 2.38i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.34 + 3.22i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.76 + 6.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.48iT - 89T^{2} \)
97 \( 1 + (-2.13 + 1.55i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.14931865234593967615156257078, −13.71240417352977232475386250243, −12.62842238015663402871392655130, −11.54771193525785857466538756999, −10.18450091798313999433975413160, −9.701092896837704843099700570076, −7.83305622286453372220932056079, −5.74253748640372184863498487386, −4.78522636852462276319893794426, −3.09194171665896979911122556877, 2.74378335413805103047111511603, 5.10945606701880016843023575444, 6.45412151110563902973231865948, 7.41148146512758260423166613723, 8.750150805400269018790441243396, 10.57938813420588654897955701940, 11.95736301832014010669538913684, 12.65461842036408814780909771927, 14.07110458261118941273089449187, 14.28963195660733860857162982352

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