↑

Properties

Label 2-63-9.2-c2-0-8
Degree $2$
Conductor $63$
Sign $0.843 - 0.536i$
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  + (2.27 + 1.31i)2-s + (2.79 − 1.08i)3-s + (1.44 + 2.49i)4-s + (−7.02 + 4.05i)5-s + (7.77 + 1.20i)6-s + (1.32 − 2.29i)7-s − 2.93i·8-s + (6.64 − 6.07i)9-s − 21.2·10-s + (−11.7 − 6.79i)11-s + (6.74 + 5.41i)12-s + (7.28 + 12.6i)13-s + (6.01 − 3.47i)14-s + (−15.2 + 18.9i)15-s + (9.61 − 16.6i)16-s + 13.5i·17-s + ⋯
L(s)  = 1  + (1.13 + 0.655i)2-s + (0.932 − 0.362i)3-s + (0.360 + 0.623i)4-s + (−1.40 + 0.811i)5-s + (1.29 + 0.200i)6-s + (0.188 − 0.327i)7-s − 0.366i·8-s + (0.737 − 0.675i)9-s − 2.12·10-s + (−1.07 − 0.617i)11-s + (0.561 + 0.451i)12-s + (0.560 + 0.971i)13-s + (0.429 − 0.247i)14-s + (−1.01 + 1.26i)15-s + (0.600 − 1.04i)16-s + 0.795i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.05064 + 0.596850i\)
\(L(\frac12)\) \(\approx\) \(2.05064 + 0.596850i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.79 + 1.08i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
good2 \( 1 + (-2.27 - 1.31i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (7.02 - 4.05i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (11.7 + 6.79i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.28 - 12.6i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 13.5iT - 289T^{2} \)
19 \( 1 + 5.74T + 361T^{2} \)
23 \( 1 + (18.2 - 10.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-42.0 - 24.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (3.97 + 6.88i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 13.3T + 1.36e3T^{2} \)
41 \( 1 + (3.28 - 1.89i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-2.48 + 4.30i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (39.4 + 22.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 46.0iT - 2.80e3T^{2} \)
59 \( 1 + (-47.0 + 27.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.542 - 0.939i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.0 - 29.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 114. iT - 5.04e3T^{2} \)
73 \( 1 + 127.T + 5.32e3T^{2} \)
79 \( 1 + (14.3 - 24.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (11.7 + 6.75i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 61.9iT - 7.92e3T^{2} \)
97 \( 1 + (-44.5 + 77.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−14.68309316959036522370826396073, −13.96856772979650138302120582348, −13.00279507578223636555349127090, −11.79507339506920203714879676523, −10.43004901097880512933450271106, −8.391766432279688571886919319020, −7.47070483080666130236520409846, −6.41784531639109641177061902549, −4.28197348766683158787866748857, −3.30277392828631297559555099881, 2.79087650131384434586300695130, 4.17171463150409717751715821262, 5.06208489273984990864360050270, 7.88004961640669578912457054289, 8.460414779601690390416190788120, 10.29558108078073146232069893760, 11.60231669803088904139279046893, 12.59854713022855214743586597563, 13.27534415574447327410732477351, 14.56249470188631780344948885332

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