↑

Properties

Label 2-63-63.25-c3-0-18
Degree $2$
Conductor $63$
Sign $0.535 + 0.844i$
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  + 3.86·2-s + (−3.99 − 3.31i)3-s + 6.95·4-s + (6.67 − 11.5i)5-s + (−15.4 − 12.8i)6-s + (15.1 − 10.7i)7-s − 4.05·8-s + (4.99 + 26.5i)9-s + (25.8 − 44.7i)10-s + (14.3 + 24.8i)11-s + (−27.8 − 23.0i)12-s + (2.75 + 4.77i)13-s + (58.4 − 41.4i)14-s + (−65.0 + 24.1i)15-s − 71.2·16-s + (−8.48 + 14.6i)17-s + ⋯
L(s)  = 1  + 1.36·2-s + (−0.769 − 0.638i)3-s + 0.868·4-s + (0.597 − 1.03i)5-s + (−1.05 − 0.872i)6-s + (0.815 − 0.578i)7-s − 0.179·8-s + (0.185 + 0.982i)9-s + (0.816 − 1.41i)10-s + (0.393 + 0.681i)11-s + (−0.668 − 0.554i)12-s + (0.0588 + 0.101i)13-s + (1.11 − 0.790i)14-s + (−1.12 + 0.415i)15-s − 1.11·16-s + (−0.120 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.10919 - 1.16014i\)
\(L(\frac12)\) \(\approx\) \(2.10919 - 1.16014i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.99 + 3.31i)T \)
7 \( 1 + (-15.1 + 10.7i)T \)
good2 \( 1 - 3.86T + 8T^{2} \)
5 \( 1 + (-6.67 + 11.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-14.3 - 24.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2.75 - 4.77i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (8.48 - 14.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-31.6 - 54.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (67.4 - 116. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-118. + 204. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 92.9T + 2.97e4T^{2} \)
37 \( 1 + (-202. - 351. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (166. + 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (173. - 299. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 304.T + 1.03e5T^{2} \)
53 \( 1 + (-332. + 575. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 137.T + 2.05e5T^{2} \)
61 \( 1 + 620.T + 2.26e5T^{2} \)
67 \( 1 + 587.T + 3.00e5T^{2} \)
71 \( 1 + 121.T + 3.57e5T^{2} \)
73 \( 1 + (-143. + 248. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 977.T + 4.93e5T^{2} \)
83 \( 1 + (507. - 879. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (258. + 448. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (823. - 1.42e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−13.76354787382097561905298557423, −13.36189523783746139049785663105, −12.17893779235628669040424641110, −11.58136172811221209188111411739, −9.872209917139881438385912662984, −8.043807015466556677052953879645, −6.45188663387159546873180621479, −5.27641971089580185288080406496, −4.39629753636378727629393624667, −1.58305558458719571831842803208, 2.93770133525388635279835071807, 4.58322846523268461782897288711, 5.74675862536324836344517080822, 6.62591209137794397774405151331, 8.981920720407758427937539562867, 10.55002356867963210277201048963, 11.42328367106373772432480434176, 12.32382289775161061434461076513, 13.82880447539317118448676163515, 14.55097163885249972165228780460

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