↑

Properties

Label 2-666-37.28-c1-0-15
Degree $2$
Conductor $666$
Sign $0.385 + 0.922i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (1.97 − 2.35i)5-s + (0.153 + 0.128i)7-s + (0.866 − 0.5i)8-s + (1.53 − 2.66i)10-s + (−2.17 − 3.76i)11-s + (−1.59 − 4.38i)13-s + (0.173 + 0.0999i)14-s + (0.766 − 0.642i)16-s + (−2.32 + 6.38i)17-s + (4.07 + 0.719i)19-s + (1.05 − 2.89i)20-s + (−2.79 − 3.32i)22-s + (0.896 + 0.517i)23-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.469 − 0.171i)4-s + (0.885 − 1.05i)5-s + (0.0578 + 0.0485i)7-s + (0.306 − 0.176i)8-s + (0.486 − 0.843i)10-s + (−0.654 − 1.13i)11-s + (−0.443 − 1.21i)13-s + (0.0462 + 0.0267i)14-s + (0.191 − 0.160i)16-s + (−0.563 + 1.54i)17-s + (0.935 + 0.165i)19-s + (0.235 − 0.647i)20-s + (−0.595 − 0.709i)22-s + (0.187 + 0.107i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.385 + 0.922i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07809 - 1.38357i\)
\(L(\frac12)\) \(\approx\) \(2.07809 - 1.38357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 \)
37 \( 1 + (-5.64 - 2.27i)T \)
good5 \( 1 + (-1.97 + 2.35i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.153 - 0.128i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (2.17 + 3.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.59 + 4.38i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.32 - 6.38i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (-4.07 - 0.719i)T + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (-0.896 - 0.517i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.25 - 0.727i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.10iT - 31T^{2} \)
41 \( 1 + (-4.46 + 1.62i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 0.399iT - 43T^{2} \)
47 \( 1 + (4.10 - 7.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.65 + 7.26i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-2.69 - 3.21i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.60 + 9.89i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-6.67 - 5.59i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.45 - 13.9i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 7.27T + 73T^{2} \)
79 \( 1 + (4.04 - 4.82i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-14.4 - 5.27i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (2.06 + 2.46i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.07 - 1.19i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43731812880103044471724441139, −9.609268082578198541354874229692, −8.527746912922825613987690275899, −7.910291773952747493762331583729, −6.45125591670938919195052784649, −5.46279539862862352150874131127, −5.22198504709994650646162620537, −3.76162750547017298626260194468, −2.58367649238597973120725074239, −1.15302491183796279839562357295, 2.16279481758778609817008404746, 2.77502915926724515891825867208, 4.34115887882032185068097180090, 5.16788891118041866544149320283, 6.25696140994017906189557709993, 7.11061866781761058476195919034, 7.54021937395779933772834745761, 9.354101566554993638643835068235, 9.725607241063525473968396491943, 10.79216370301756110761572779869

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