↑

Properties

Label 2-666-111.29-c1-0-4
Degree $2$
Conductor $666$
Sign $0.557 - 0.830i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.765 + 0.205i)5-s + (−0.896 + 1.55i)7-s + (0.707 + 0.707i)8-s − 0.792·10-s + 4.98·11-s + (0.0278 + 0.103i)13-s + (−1.26 + 1.26i)14-s + (0.500 + 0.866i)16-s + (0.912 − 3.40i)17-s + (2.21 + 8.26i)19-s + (−0.765 − 0.205i)20-s + (4.81 + 1.28i)22-s + (5.99 + 5.99i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.342 + 0.0917i)5-s + (−0.338 + 0.586i)7-s + (0.249 + 0.249i)8-s − 0.250·10-s + 1.50·11-s + (0.00771 + 0.0288i)13-s + (−0.338 + 0.338i)14-s + (0.125 + 0.216i)16-s + (0.221 − 0.825i)17-s + (0.507 + 1.89i)19-s + (−0.171 − 0.0458i)20-s + (1.02 + 0.275i)22-s + (1.25 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92731 + 1.02717i\)
\(L(\frac12)\) \(\approx\) \(1.92731 + 1.02717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
37 \( 1 + (-4.26 - 4.33i)T \)
good5 \( 1 + (0.765 - 0.205i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.896 - 1.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 + (-0.0278 - 0.103i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.912 + 3.40i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.21 - 8.26i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.99 - 5.99i)T + 23iT^{2} \)
29 \( 1 + (4.02 - 4.02i)T - 29iT^{2} \)
31 \( 1 + (6.76 + 6.76i)T + 31iT^{2} \)
41 \( 1 + (-4.52 + 7.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 2.24i)T - 43iT^{2} \)
47 \( 1 + 8.63iT - 47T^{2} \)
53 \( 1 + (-0.970 + 0.560i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.06 + 11.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.05 - 0.549i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.9 + 7.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.01 + 2.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.90iT - 73T^{2} \)
79 \( 1 + (0.0278 + 0.103i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.41 + 1.97i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.75 - 1.54i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.841 - 0.841i)T - 97iT^{2} \)
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

−10.92635990935726329139910381951, −9.529118270768441591721741249324, −9.142737966362725900908776935059, −7.74190367023563840893416391986, −7.13204138174596516052886486685, −5.98122147246775647745364185880, −5.37713459225581340258012708977, −3.90767717392588693936686499244, −3.35818687986958018913227119675, −1.69942286776733429500690181661, 1.07378343520189929183657858513, 2.79366274510797811979155525608, 3.94572965042898670687287799003, 4.56055734831811470951948812019, 5.91299960254646780814880485368, 6.78387764978820352219689915123, 7.46369818383178640937002186020, 8.835811690246319518559030393690, 9.477160208422554731200848495729, 10.66766100403353724818074236569

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