↑

Properties

Label 2-63-21.5-c7-0-14
Degree $2$
Conductor $63$
Sign $-0.715 - 0.698i$
Analytic cond. $19.6802$
Root an. cond. $4.43624$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.8 − 6.86i)2-s + (30.2 + 52.3i)4-s + (99.9 − 173. i)5-s + (−671. − 610. i)7-s + 927. i·8-s + (−2.37e3 + 1.37e3i)10-s + (4.94e3 − 2.85e3i)11-s − 1.05e4i·13-s + (3.78e3 + 1.18e4i)14-s + (1.02e4 − 1.77e4i)16-s + (−7.19e3 − 1.24e4i)17-s + (−3.28e4 − 1.89e4i)19-s + 1.20e4·20-s − 7.83e4·22-s + (4.61e4 + 2.66e4i)23-s + ⋯
L(s)  = 1  + (−1.05 − 0.606i)2-s + (0.236 + 0.408i)4-s + (0.357 − 0.619i)5-s + (−0.739 − 0.672i)7-s + 0.640i·8-s + (−0.751 + 0.433i)10-s + (1.11 − 0.646i)11-s − 1.33i·13-s + (0.369 + 1.15i)14-s + (0.624 − 1.08i)16-s + (−0.355 − 0.614i)17-s + (−1.09 − 0.633i)19-s + 0.337·20-s − 1.56·22-s + (0.790 + 0.456i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(19.6802\)
Root analytic conductor: \(4.43624\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :7/2),\ -0.715 - 0.698i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.179940 + 0.442174i\)
\(L(\frac12)\) \(\approx\) \(0.179940 + 0.442174i\)
\(L(\frac{9}{2})\) 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 \( 1 + (671. + 610. i)T \)
good2 \( 1 + (11.8 + 6.86i)T + (64 + 110. i)T^{2} \)
5 \( 1 + (-99.9 + 173. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-4.94e3 + 2.85e3i)T + (9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 1.05e4iT - 6.27e7T^{2} \)
17 \( 1 + (7.19e3 + 1.24e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (3.28e4 + 1.89e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-4.61e4 - 2.66e4i)T + (1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.11e5iT - 1.72e10T^{2} \)
31 \( 1 + (1.34e5 - 7.75e4i)T + (1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (2.03e4 - 3.51e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 6.93e5T + 1.94e11T^{2} \)
43 \( 1 + 5.75e5T + 2.71e11T^{2} \)
47 \( 1 + (1.15e5 - 2.00e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-9.93e5 + 5.73e5i)T + (5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-2.38e5 - 4.13e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (9.92e5 + 5.72e5i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.88e6 - 3.26e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 5.03e6iT - 9.09e12T^{2} \)
73 \( 1 + (-1.14e6 + 6.60e5i)T + (5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (1.53e6 - 2.65e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 4.95e5T + 2.71e13T^{2} \)
89 \( 1 + (4.05e6 - 7.03e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.70e7iT - 8.07e13T^{2} \)
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

−12.74859425191714504198967526743, −11.29609771709750661768785280759, −10.37696900048091853006332908133, −9.283039449944762761114370174157, −8.558983561890886959532142933854, −6.88460465823449833761705979462, −5.22712583166778177672640540957, −3.22653566618497816251161603372, −1.31257032282796440128601838884, −0.26509394077804247633628127983, 1.93318910709638378530911475397, 3.96061207547006227159832935400, 6.40782712012112488895140267907, 6.79198910698315066558992052054, 8.568717859674796547599832081337, 9.367562793828724045819129595282, 10.32440115458052652220661290023, 11.90441773983100487957533466976, 13.04206132069980407598639978621, 14.56086628143917823863940553065

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