Properties

Label 2-63-63.59-c3-0-11
Degree $2$
Conductor $63$
Sign $0.222 - 0.974i$
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  + (3.68 + 2.12i)2-s + (3.24 + 4.05i)3-s + (5.03 + 8.72i)4-s − 2.97·5-s + (3.31 + 21.8i)6-s + (−4.05 − 18.0i)7-s + 8.83i·8-s + (−5.95 + 26.3i)9-s + (−10.9 − 6.32i)10-s − 33.6i·11-s + (−19.0 + 48.7i)12-s + (22.0 + 12.7i)13-s + (23.5 − 75.1i)14-s + (−9.65 − 12.0i)15-s + (21.5 − 37.2i)16-s + (−55.6 + 96.3i)17-s + ⋯
L(s)  = 1  + (1.30 + 0.751i)2-s + (0.624 + 0.781i)3-s + (0.629 + 1.09i)4-s − 0.266·5-s + (0.225 + 1.48i)6-s + (−0.218 − 0.975i)7-s + 0.390i·8-s + (−0.220 + 0.975i)9-s + (−0.346 − 0.200i)10-s − 0.922i·11-s + (−0.459 + 1.17i)12-s + (0.471 + 0.272i)13-s + (0.448 − 1.43i)14-s + (−0.166 − 0.207i)15-s + (0.336 − 0.582i)16-s + (−0.793 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.222 - 0.974i$
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: \((2,\ 63,\ (\ :3/2),\ 0.222 - 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.31306 + 1.84497i\)
\(L(\frac12)\) \(\approx\) \(2.31306 + 1.84497i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.24 - 4.05i)T \)
7 \( 1 + (4.05 + 18.0i)T \)
good2 \( 1 + (-3.68 - 2.12i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 2.97T + 125T^{2} \)
11 \( 1 + 33.6iT - 1.33e3T^{2} \)
13 \( 1 + (-22.0 - 12.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (55.6 - 96.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-66.6 + 38.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 27.8iT - 1.21e4T^{2} \)
29 \( 1 + (-87.6 + 50.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (57.9 - 33.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (146. + 254. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (62.2 - 107. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (151. + 262. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (183. - 317. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-281. - 162. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-388. - 673. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-162. - 93.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-358. - 621. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 762. iT - 3.57e5T^{2} \)
73 \( 1 + (409. + 236. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-353. + 612. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (406. + 703. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (393. + 681. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.51e3 + 871. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−14.60151139619921358418846924390, −13.72380324058515804808206024847, −13.17451166989000174030875477961, −11.40560198984661482641360687700, −10.20785033526990448527580826334, −8.611128831396765548258991580528, −7.28658232780094194316981556250, −5.83467336595579825152961304816, −4.27992167093098595091767216981, −3.48392910824633422160436136895, 2.12140300250089151501913477341, 3.38448974804409015206317119578, 5.13582352205223500773765591578, 6.65802189840201357333168887804, 8.254902285179642569409622194984, 9.666856590361454542336979808537, 11.56271567670662796113881433720, 12.13198755550789361851161766234, 13.09383313771373254640094307053, 13.91704619759994895475618453693

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