↑

Properties

Label 2-63-63.20-c3-0-12
Degree $2$
Conductor $63$
Sign $0.246 - 0.969i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.27 + 2.46i)2-s + (3.18 + 4.10i)3-s + (8.16 + 14.1i)4-s + (−7.65 − 13.2i)5-s + (3.49 + 25.3i)6-s + (6.32 − 17.4i)7-s + 41.0i·8-s + (−6.67 + 26.1i)9-s − 75.5i·10-s + (−36.9 − 21.3i)11-s + (−31.9 + 78.5i)12-s + (−24.2 + 14.0i)13-s + (69.9 − 58.7i)14-s + (30.0 − 73.6i)15-s + (−35.9 + 62.2i)16-s + 82.0·17-s + ⋯
L(s)  = 1  + (1.51 + 0.871i)2-s + (0.613 + 0.789i)3-s + (1.02 + 1.76i)4-s + (−0.684 − 1.18i)5-s + (0.237 + 1.72i)6-s + (0.341 − 0.939i)7-s + 1.81i·8-s + (−0.247 + 0.968i)9-s − 2.38i·10-s + (−1.01 − 0.584i)11-s + (−0.769 + 1.89i)12-s + (−0.517 + 0.299i)13-s + (1.33 − 1.12i)14-s + (0.516 − 1.26i)15-s + (−0.562 + 0.973i)16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.53159 + 1.96758i\)
\(L(\frac12)\) \(\approx\) \(2.53159 + 1.96758i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.18 - 4.10i)T \)
7 \( 1 + (-6.32 + 17.4i)T \)
good2 \( 1 + (-4.27 - 2.46i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (7.65 + 13.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (36.9 + 21.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (24.2 - 14.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 82.0T + 4.91e3T^{2} \)
19 \( 1 - 113. iT - 6.85e3T^{2} \)
23 \( 1 + (25.2 - 14.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-16.5 - 9.56i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-97.0 + 56.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 69.2T + 5.06e4T^{2} \)
41 \( 1 + (242. + 419. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (194. - 337. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (53.4 - 92.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 207. iT - 1.48e5T^{2} \)
59 \( 1 + (97.8 + 169. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-446. - 257. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 640. iT - 3.57e5T^{2} \)
73 \( 1 - 528. iT - 3.89e5T^{2} \)
79 \( 1 + (-20.7 + 35.8i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-554. + 960. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 121.T + 7.04e5T^{2} \)
97 \( 1 + (748. + 432. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−14.58536866046056147709149164856, −13.84911243637682654361469308940, −12.86499899326778753887656249972, −11.77472829344099859180158215535, −10.15474497305265970321054275773, −8.213072496123444055009123351854, −7.67248695282968188025704180557, −5.45894958243317149247799773434, −4.53100212275251133195691764556, −3.51859767651720615352657556582, 2.39772430761482937663789469610, 3.19420344811809639697920340278, 5.13351565353588896926278338875, 6.70918096794893541626353557288, 7.981981725164472331366883381970, 10.07837631070928926063557447980, 11.39578989620512365444664474054, 12.12998033055051110893924763665, 13.03380561006839046430274092502, 14.15429358515862637545898802307

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