Properties

Label 2-63-63.4-c3-0-5
Degree $2$
Conductor $63$
Sign $-0.185 - 0.982i$
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  + (−1.83 + 3.17i)2-s + (−2.73 − 4.41i)3-s + (−2.73 − 4.73i)4-s + 14.7·5-s + (19.0 − 0.597i)6-s + (0.242 + 18.5i)7-s − 9.29·8-s + (−12.0 + 24.1i)9-s + (−27.1 + 46.9i)10-s + 48.5·11-s + (−13.4 + 25.0i)12-s + (−33.6 + 58.1i)13-s + (−59.2 − 33.2i)14-s + (−40.4 − 65.2i)15-s + (38.9 − 67.4i)16-s + (−5.40 + 9.36i)17-s + ⋯
L(s)  = 1  + (−0.648 + 1.12i)2-s + (−0.526 − 0.849i)3-s + (−0.341 − 0.591i)4-s + 1.32·5-s + (1.29 − 0.0406i)6-s + (0.0130 + 0.999i)7-s − 0.410·8-s + (−0.444 + 0.895i)9-s + (−0.857 + 1.48i)10-s + 1.33·11-s + (−0.323 + 0.602i)12-s + (−0.716 + 1.24i)13-s + (−1.13 − 0.633i)14-s + (−0.696 − 1.12i)15-s + (0.608 − 1.05i)16-s + (−0.0771 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.185 - 0.982i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.185 - 0.982i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.649057 + 0.783278i\)
\(L(\frac12)\) \(\approx\) \(0.649057 + 0.783278i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.73 + 4.41i)T \)
7 \( 1 + (-0.242 - 18.5i)T \)
good2 \( 1 + (1.83 - 3.17i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 14.7T + 125T^{2} \)
11 \( 1 - 48.5T + 1.33e3T^{2} \)
13 \( 1 + (33.6 - 58.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (5.40 - 9.36i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-67.2 - 116. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 84.3T + 1.21e4T^{2} \)
29 \( 1 + (55.1 + 95.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (75.5 + 130. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (152. + 263. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-127. + 220. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-41.3 - 71.5i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (23.0 - 39.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (3.20 - 5.55i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (5.59 + 9.69i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (136. - 235. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-28.7 - 49.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 521.T + 3.57e5T^{2} \)
73 \( 1 + (189. - 327. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-472. + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (411. + 711. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (12.4 + 21.6i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-22.7 - 39.3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−14.65605254294617741816330037097, −14.02678821886200025390584290683, −12.46892176452653414033516442488, −11.64788815275029106364324246679, −9.613679327159989991881767246623, −8.915816224071146389539705733748, −7.34550102362484662083257173123, −6.26785025600922560153252191353, −5.60065662414689359730453614354, −1.91667591897777742908354663273, 1.00304732156944939400867448509, 3.18925603005932284355060513845, 5.13914067267816053718146956762, 6.66901556328650439598449941065, 9.103382298374001392161040489695, 9.757930736707775419064460666606, 10.56937787233956973497250822161, 11.44740906191105981915114366319, 12.76789641788959713800703130420, 14.08488957369833522078154843680

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