Properties

Label 2-63-7.3-c6-0-4
Degree $2$
Conductor $63$
Sign $0.605 + 0.795i$
Analytic cond. $14.4934$
Root an. cond. $3.80702$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−6 − 10.3i)2-s + (−40 + 69.2i)4-s + (−157.5 + 90.9i)5-s − 343·7-s + 192.·8-s + (1.89e3 + 1.09e3i)10-s + (739.5 − 1.28e3i)11-s + 484. i·13-s + (2.05e3 + 3.56e3i)14-s + (1.40e3 + 2.43e3i)16-s + (2.61e3 + 1.50e3i)17-s + (5.95e3 − 3.43e3i)19-s − 1.45e4i·20-s − 1.77e4·22-s + (−2.95e3 − 5.12e3i)23-s + ⋯
L(s)  = 1  + (−0.750 − 1.29i)2-s + (−0.625 + 1.08i)4-s + (−1.26 + 0.727i)5-s − 7-s + 0.375·8-s + (1.89 + 1.09i)10-s + (0.555 − 0.962i)11-s + 0.220i·13-s + (0.750 + 1.29i)14-s + (0.343 + 0.595i)16-s + (0.532 + 0.307i)17-s + (0.867 − 0.501i)19-s − 1.81i·20-s − 1.66·22-s + (−0.242 − 0.420i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(14.4934\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.565253 - 0.280211i\)
\(L(\frac12)\) \(\approx\) \(0.565253 - 0.280211i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 343T \)
good2 \( 1 + (6 + 10.3i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (157.5 - 90.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-739.5 + 1.28e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 484. iT - 4.82e6T^{2} \)
17 \( 1 + (-2.61e3 - 1.50e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-5.95e3 + 3.43e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (2.95e3 + 5.12e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 3.97e3T + 5.94e8T^{2} \)
31 \( 1 + (1.10e4 + 6.40e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-3.07e4 - 5.33e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.10e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.74e4T + 6.32e9T^{2} \)
47 \( 1 + (-2.65e4 + 1.53e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (3.02e4 - 5.24e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.86e5 - 1.07e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.40e5 + 8.13e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.34e5 + 2.32e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 1.01e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.75e5 - 1.58e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.81e5 + 3.13e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 2.16e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.15e6 + 6.67e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.51e6iT - 8.32e11T^{2} \)
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

−13.17006697111538048089198297608, −11.89708250708182099840510529197, −11.37325435662897219311907348316, −10.27160597846059796390011380630, −9.171543571106060066802124567160, −7.922100225694134851695399254684, −6.42171832856460709226162068612, −3.73267682614220158575535845778, −2.96684403353166086335265013419, −0.73485743619374839299270418305, 0.56749687417866988727752900766, 3.75921605741168136011107076043, 5.47905694294126829380028430448, 7.02434071419374814316981860964, 7.76738744420237876769704572930, 8.998473721598464354593354465028, 9.875469964125696764396769346250, 11.86613171560798112239510907754, 12.64373345049078606426795020306, 14.34963545245706284224031292325

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