Properties

Label 2-63-63.2-c2-0-5
Degree $2$
Conductor $63$
Sign $0.748 - 0.663i$
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  + (−0.444 + 0.256i)2-s + (2.83 − 0.970i)3-s + (−1.86 + 3.23i)4-s + 7.02i·5-s + (−1.01 + 1.16i)6-s + (5.34 − 4.51i)7-s − 3.97i·8-s + (7.11 − 5.50i)9-s + (−1.80 − 3.12i)10-s + 3.64i·11-s + (−2.16 + 10.9i)12-s + (−3.79 − 6.56i)13-s + (−1.21 + 3.38i)14-s + (6.81 + 19.9i)15-s + (−6.45 − 11.1i)16-s + (−17.5 + 10.1i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.128i)2-s + (0.946 − 0.323i)3-s + (−0.467 + 0.808i)4-s + 1.40i·5-s + (−0.168 + 0.193i)6-s + (0.763 − 0.645i)7-s − 0.496i·8-s + (0.790 − 0.612i)9-s + (−0.180 − 0.312i)10-s + 0.331i·11-s + (−0.180 + 0.916i)12-s + (−0.291 − 0.505i)13-s + (−0.0869 + 0.241i)14-s + (0.454 + 1.32i)15-s + (−0.403 − 0.698i)16-s + (−1.03 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.748 - 0.663i$
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: \((2,\ 63,\ (\ :1),\ 0.748 - 0.663i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24822 + 0.473764i\)
\(L(\frac12)\) \(\approx\) \(1.24822 + 0.473764i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.83 + 0.970i)T \)
7 \( 1 + (-5.34 + 4.51i)T \)
good2 \( 1 + (0.444 - 0.256i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 7.02iT - 25T^{2} \)
11 \( 1 - 3.64iT - 121T^{2} \)
13 \( 1 + (3.79 + 6.56i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (17.5 - 10.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.6 + 23.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + 3.94iT - 529T^{2} \)
29 \( 1 + (23.7 + 13.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (2.42 - 4.20i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (18.7 - 32.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-61.1 + 35.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.41 - 16.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-20.7 + 11.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (23.1 - 13.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (45.7 + 26.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-53.4 - 92.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.0 - 88.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 138. iT - 5.04e3T^{2} \)
73 \( 1 + (34.7 + 60.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (11.6 + 20.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (25.9 + 14.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (135. + 78.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-2.93 + 5.07i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−14.73230190774155737642473721313, −13.80638496103621959105062192892, −12.96357923264578454496620660907, −11.41001743130091250308917191128, −10.15853692528926140866892825319, −8.811314708680113552694953374459, −7.56492788635381252845259553033, −6.99761301894391833859723525374, −4.18008418594566460451387756612, −2.74560800501549119188117556089, 1.73456326330976406593707941239, 4.44550911622443695514321555456, 5.43726273485096605630592272746, 7.970104274273986243816075502513, 9.008829907625090423661554133791, 9.456371406957569172872052061019, 11.07525480227897164268097942618, 12.50559721831893263162009548231, 13.72362949360698467989294192926, 14.48031092818660588938252194852

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