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Properties

Label 2-66-33.29-c1-0-3
Degree $2$
Conductor $66$
Sign $0.556 + 0.830i$
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 + (1.25 − 1.19i)3-s + (0.309 + 0.951i)4-s + (−0.442 − 0.609i)5-s + (−1.71 + 0.232i)6-s + (−0.442 + 0.143i)7-s + (0.309 − 0.951i)8-s + (0.133 − 2.99i)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 + (−1.28 − 0.232i)15-s + (−0.809 + 0.587i)16-s + (−2.99 + 2.17i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.722 − 0.691i)3-s + (0.154 + 0.475i)4-s + (−0.197 − 0.272i)5-s + (−0.700 + 0.0950i)6-s + (−0.167 + 0.0543i)7-s + (0.109 − 0.336i)8-s + (0.0445 − 0.999i)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 + (−0.331 − 0.0600i)15-s + (−0.202 + 0.146i)16-s + (−0.726 + 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723811 - 0.386273i\)
\(L(\frac12)\) \(\approx\) \(0.723811 - 0.386273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 1.19i)T \)
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

−14.58455401121154082218733484931, −13.50164486335791337930569255554, −12.30074854806150342705677271106, −11.64268384177504019286407389414, −9.752194215935985454467491864550, −9.023450608010200502969407141658, −7.71354821013588223397333980671, −6.64543745486974754542462055427, −4.05216809851475919153278365587, −2.03520195845549829064966717141, 3.09880188200855684634985287238, 5.03771162290902244135989256353, 6.89138442137476611124330966393, 8.181918584120871000769405975707, 9.209866538493429171187771429470, 10.26886621296724380584709766928, 11.32058490802544789379424567193, 13.07686830837794254921162718986, 14.38947854052996128579277217600, 15.02151717832383531534504776874

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