↑

Properties

Label 2-63-63.2-c2-0-9
Degree $2$
Conductor $63$
Sign $-0.114 + 0.993i$
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  + (−1.11 + 0.645i)2-s + (−0.401 − 2.97i)3-s + (−1.16 + 2.02i)4-s − 4.83i·5-s + (2.36 + 3.06i)6-s + (−1.74 − 6.78i)7-s − 8.17i·8-s + (−8.67 + 2.38i)9-s + (3.12 + 5.40i)10-s − 13.6i·11-s + (6.47 + 2.65i)12-s + (8.59 + 14.8i)13-s + (6.32 + 6.45i)14-s + (−14.3 + 1.94i)15-s + (0.609 + 1.05i)16-s + (7.11 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.558 + 0.322i)2-s + (−0.133 − 0.990i)3-s + (−0.291 + 0.505i)4-s − 0.967i·5-s + (0.394 + 0.510i)6-s + (−0.248 − 0.968i)7-s − 1.02i·8-s + (−0.964 + 0.265i)9-s + (0.312 + 0.540i)10-s − 1.24i·11-s + (0.539 + 0.221i)12-s + (0.661 + 1.14i)13-s + (0.451 + 0.461i)14-s + (−0.958 + 0.129i)15-s + (0.0381 + 0.0660i)16-s + (0.418 − 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 63,\ (\ :1),\ -0.114 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.457906 - 0.513747i\)
\(L(\frac12)\) \(\approx\) \(0.457906 - 0.513747i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.401 + 2.97i)T \)
7 \( 1 + (1.74 + 6.78i)T \)
good2 \( 1 + (1.11 - 0.645i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 4.83iT - 25T^{2} \)
11 \( 1 + 13.6iT - 121T^{2} \)
13 \( 1 + (-8.59 - 14.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-7.11 + 4.10i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.93 - 12.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 18.5iT - 529T^{2} \)
29 \( 1 + (18.1 + 10.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-25.7 + 44.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-13.8 + 23.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-18.4 + 10.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 + 31.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-51.4 + 29.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (81.0 - 46.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.45 + 0.838i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32.7 - 56.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-27.6 + 47.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 14.2iT - 5.04e3T^{2} \)
73 \( 1 + (-32.6 - 56.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-3.35 - 5.80i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-11.6 - 6.69i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (13.7 + 7.92i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (40.7 - 70.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−13.88871437479706403680959903846, −13.41040730439188672028087401068, −12.36908750732045298355874115895, −11.18709454170994439026370985464, −9.350103214399215099137065272314, −8.377502267583545749969419152452, −7.42910786200716353808988303314, −6.05868731829985266932846704086, −3.91649967167577919798283582614, −0.820930587439350837760609944718, 2.83231441187766665421258367911, 4.94911417053618206468802881846, 6.26465055781148643017928225130, 8.381403432856388210396206169256, 9.526291891845100989965199325178, 10.38849368588522712360870523616, 11.12651484322886917746104674989, 12.59280318195281690411015037239, 14.42163886995742410432693518323, 15.01743796922552240804097610311

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