Properties

Label 2-552-184.85-c1-0-19
Degree $2$
Conductor $552$
Sign $0.860 - 0.509i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  + (−1.08 − 0.910i)2-s + (0.540 + 0.841i)3-s + (0.343 + 1.97i)4-s + (−2.04 + 0.936i)5-s + (0.180 − 1.40i)6-s + (3.54 − 1.04i)7-s + (1.42 − 2.44i)8-s + (−0.415 + 0.909i)9-s + (3.07 + 0.852i)10-s + (2.25 − 1.95i)11-s + (−1.47 + 1.35i)12-s + (−1.41 + 4.83i)13-s + (−4.78 − 2.09i)14-s + (−1.89 − 1.21i)15-s + (−3.76 + 1.35i)16-s + (0.385 − 2.68i)17-s + ⋯
L(s)  = 1  + (−0.765 − 0.643i)2-s + (0.312 + 0.485i)3-s + (0.171 + 0.985i)4-s + (−0.916 + 0.418i)5-s + (0.0736 − 0.572i)6-s + (1.33 − 0.393i)7-s + (0.502 − 0.864i)8-s + (−0.138 + 0.303i)9-s + (0.970 + 0.269i)10-s + (0.679 − 0.588i)11-s + (−0.424 + 0.390i)12-s + (−0.393 + 1.34i)13-s + (−1.27 − 0.560i)14-s + (−0.489 − 0.314i)15-s + (−0.940 + 0.338i)16-s + (0.0934 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04031 + 0.284740i\)
\(L(\frac12)\) \(\approx\) \(1.04031 + 0.284740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.910i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (-2.30 - 4.20i)T \)
good5 \( 1 + (2.04 - 0.936i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-3.54 + 1.04i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-2.25 + 1.95i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (1.41 - 4.83i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.385 + 2.68i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.208 + 0.0300i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (1.25 + 0.180i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-7.50 - 4.82i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-8.01 - 3.65i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (-2.68 - 5.87i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (1.18 + 1.84i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 + (-1.35 - 4.59i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (-0.831 + 2.83i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-2.51 + 3.91i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-9.06 - 7.85i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (9.89 - 11.4i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (2.26 + 15.7i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-2.60 - 0.763i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-1.34 - 0.615i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-6.87 + 4.41i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (3.69 + 8.10i)T + (-63.5 + 73.3i)T^{2} \)
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

−11.13711935535124291653095740590, −9.941198252640864645725524144570, −9.150698058688334136036671665756, −8.264829653372808280419550844316, −7.60829767171640494347209401124, −6.74298974730196724342407579756, −4.74757179481762902055556689353, −4.03865586374143796077980068690, −2.96073734540135598127937360256, −1.41117924501819443771229696064, 0.907027241019118378349660073073, 2.30961488454738329472055999237, 4.26954395404060511874040523759, 5.20331420773402598344736687716, 6.34115764949662679103597089011, 7.54028543141654937915282349368, 8.051496840477173952685130956858, 8.532228160420720366312441467884, 9.603015570711525682158466148663, 10.66683622012466178555196745005

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