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Properties

Label 2-66-33.32-c1-0-0
Degree $2$
Conductor $66$
Sign $0.174 - 0.984i$
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  − 2-s + (−1 + 1.41i)3-s + 4-s + 1.41i·5-s + (1 − 1.41i)6-s + 4.24i·7-s − 8-s + (−1.00 − 2.82i)9-s − 1.41i·10-s + (3 − 1.41i)11-s + (−1 + 1.41i)12-s − 4.24i·13-s − 4.24i·14-s + (−2.00 − 1.41i)15-s + 16-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 + 0.816i)3-s + 0.5·4-s + 0.632i·5-s + (0.408 − 0.577i)6-s + 1.60i·7-s − 0.353·8-s + (−0.333 − 0.942i)9-s − 0.447i·10-s + (0.904 − 0.426i)11-s + (−0.288 + 0.408i)12-s − 1.17i·13-s − 1.13i·14-s + (−0.516 − 0.365i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.174 - 0.984i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.444607 + 0.372904i\)
\(L(\frac12)\) \(\approx\) \(0.444607 + 0.372904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1 - 1.41i)T \)
11 \( 1 + (-3 + 1.41i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 9.89iT - 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 4.24iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−15.30556197279685379006222456182, −14.54992152661329495575061204561, −12.43098421148282694170278084302, −11.54499026212327286047161335028, −10.56781245943628211088939549144, −9.409888699472237083423453301165, −8.460277818044254713677999124030, −6.50789825526095172594949771550, −5.45095385276715594437220529329, −3.10679946474497838366817225872, 1.29388794649529126458273631718, 4.47633725444255445471405884856, 6.56861176456089163521667146027, 7.32114391188213531162876555526, 8.723552398928833326275274416666, 10.12875466226042731418245521379, 11.27519791264757477370321920144, 12.25214400689579708503436217425, 13.44543812407287555328246665904, 14.41827408583761118588371976444

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