Properties

Label 2-2664-2664.787-c0-0-1
Degree $2$
Conductor $2664$
Sign $-0.200 + 0.979i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2i·5-s − 6-s + (−0.866 − 0.5i)7-s + 8-s + 9-s − 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (−0.866 − 0.5i)14-s + 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 2i·5-s − 6-s + (−0.866 − 0.5i)7-s + 8-s + 9-s − 2i·10-s + (0.5 + 0.866i)11-s − 12-s + (−0.866 − 0.5i)14-s + 2i·15-s + 16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ -0.200 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564679038\)
\(L(\frac12)\) \(\approx\) \(1.564679038\)
\(L(1)\) 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 \( 1 + T \)
37 \( 1 + iT \)
good5 \( 1 + 2iT - T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.037422619907504946657027986837, −7.77145579924915727883509327221, −7.04296381733106549505123624695, −6.41775061277426292130878226384, −5.36309186035979060361378040275, −4.99113324398102047522602570273, −4.34037099670014904594383906227, −3.52570251681297657743600626495, −1.85165829563059512989373076064, −0.843005525993512904857495906218, 1.84583136229256293918353597142, 3.07130463288313541882682440358, 3.46753734404047600455479253084, 4.45381411582690213378896853060, 5.85585940927029425603988416484, 6.13363651756931150912699110875, 6.51212015277502486517550918827, 7.31732566697193316398329512936, 8.162588599790895881317736912630, 9.734790136347096479781064075948

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