↑

Properties

Label 2-63-63.2-c2-0-3
Degree $2$
Conductor $63$
Sign $-0.348 - 0.937i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 + 0.732i)2-s + (1.95 + 2.27i)3-s + (−0.926 + 1.60i)4-s − 1.15i·5-s + (−4.14 − 1.45i)6-s + (−2.90 + 6.36i)7-s − 8.57i·8-s + (−1.35 + 8.89i)9-s + (0.844 + 1.46i)10-s + 0.241i·11-s + (−5.46 + 1.02i)12-s + (7.70 + 13.3i)13-s + (−0.973 − 10.2i)14-s + (2.62 − 2.25i)15-s + (2.58 + 4.46i)16-s + (10.9 − 6.29i)17-s + ⋯
L(s)  = 1  + (−0.634 + 0.366i)2-s + (0.651 + 0.758i)3-s + (−0.231 + 0.401i)4-s − 0.230i·5-s + (−0.691 − 0.242i)6-s + (−0.415 + 0.909i)7-s − 1.07i·8-s + (−0.151 + 0.988i)9-s + (0.0844 + 0.146i)10-s + 0.0219i·11-s + (−0.455 + 0.0856i)12-s + (0.592 + 1.02i)13-s + (−0.0695 − 0.729i)14-s + (0.174 − 0.150i)15-s + (0.161 + 0.279i)16-s + (0.641 − 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.540432 + 0.777382i\)
\(L(\frac12)\) \(\approx\) \(0.540432 + 0.777382i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.95 - 2.27i)T \)
7 \( 1 + (2.90 - 6.36i)T \)
good2 \( 1 + (1.26 - 0.732i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 1.15iT - 25T^{2} \)
11 \( 1 - 0.241iT - 121T^{2} \)
13 \( 1 + (-7.70 - 13.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-10.9 + 6.29i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.7 + 23.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + 20.3iT - 529T^{2} \)
29 \( 1 + (-16.9 - 9.76i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.36 + 2.36i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (11.4 - 19.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (61.9 - 35.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-14.8 + 25.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.9 - 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-90.5 + 52.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (75.0 + 43.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.94 + 15.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.02 + 15.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 74.2iT - 5.04e3T^{2} \)
73 \( 1 + (-13.0 - 22.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (11.2 + 19.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (77.0 + 44.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (49.7 + 28.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-73.9 + 128. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−15.39268049899782131159460736566, −14.06159161985646001264264934452, −12.97050344670469169720377728533, −11.66339130842067550850040493853, −9.986397549623856103029593038369, −9.020188912792509877560655069426, −8.465401160847779406732741709714, −6.84908737272666203377445601485, −4.81525605413434879799002545320, −3.13659597479468712561196055934, 1.16711358625221229611121247049, 3.37493105431581895781795235795, 5.85143152917893002667215211767, 7.45478514459731146509260903206, 8.482543321298624162912978980812, 9.834187781728132216622787700637, 10.64136282001767299237040297767, 12.17503702066219294148414583337, 13.46528360312755288033056072318, 14.15939961101161717997007941389

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