Properties

Label 2-63-63.59-c3-0-13
Degree $2$
Conductor $63$
Sign $0.00668 + 0.999i$
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  + (−0.223 − 0.128i)2-s + (−3.39 + 3.93i)3-s + (−3.96 − 6.87i)4-s + 6.38·5-s + (1.26 − 0.439i)6-s + (−1.54 − 18.4i)7-s + 4.10i·8-s + (−3.91 − 26.7i)9-s + (−1.42 − 0.822i)10-s − 61.0i·11-s + (40.4 + 7.74i)12-s + (8.78 + 5.07i)13-s + (−2.03 + 4.31i)14-s + (−21.7 + 25.1i)15-s + (−31.2 + 54.0i)16-s + (22.5 − 38.9i)17-s + ⋯
L(s)  = 1  + (−0.0788 − 0.0455i)2-s + (−0.653 + 0.756i)3-s + (−0.495 − 0.858i)4-s + 0.571·5-s + (0.0860 − 0.0299i)6-s + (−0.0833 − 0.996i)7-s + 0.181i·8-s + (−0.145 − 0.989i)9-s + (−0.0450 − 0.0260i)10-s − 1.67i·11-s + (0.974 + 0.186i)12-s + (0.187 + 0.108i)13-s + (−0.0388 + 0.0824i)14-s + (−0.373 + 0.432i)15-s + (−0.487 + 0.844i)16-s + (0.321 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00668 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00668 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.00668 + 0.999i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.00668 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.617172 - 0.613061i\)
\(L(\frac12)\) \(\approx\) \(0.617172 - 0.613061i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.39 - 3.93i)T \)
7 \( 1 + (1.54 + 18.4i)T \)
good2 \( 1 + (0.223 + 0.128i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 6.38T + 125T^{2} \)
11 \( 1 + 61.0iT - 1.33e3T^{2} \)
13 \( 1 + (-8.78 - 5.07i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-22.5 + 38.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (69.6 - 40.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 27.6iT - 1.21e4T^{2} \)
29 \( 1 + (-48.9 + 28.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-92.1 + 53.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-95.6 - 165. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (15.0 - 26.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (185. + 320. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-248. + 429. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-601. - 347. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (317. + 550. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-647. - 373. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (82.3 + 142. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 278. iT - 3.57e5T^{2} \)
73 \( 1 + (313. + 181. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (278. - 482. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-514. - 891. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-730. - 1.26e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-878. + 507. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−14.03293663262856994487848430818, −13.48499398189903570731933899395, −11.55339938890241726075725234726, −10.54162841227796519209826484378, −9.886413495407116846236310379304, −8.620912316226901558314912574864, −6.37553966488659732856211469071, −5.43731655381384875575586498718, −3.92010774182211241238027663203, −0.68424732151455263565558833006, 2.24659717529494176936973967037, 4.73239733889858873525977258234, 6.21382560512852706237443107478, 7.53227153235281712342405053867, 8.783799697568368278687274476440, 10.09211091232761560636759766941, 11.77143022953001475368858181825, 12.60560071758824302434475005135, 13.18147610094428538518520168947, 14.65707386273317244932547839086

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