↑

Properties

Label 2-63-7.2-c3-0-2
Degree $2$
Conductor $63$
Sign $0.0306 - 0.999i$
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  + (0.124 − 0.214i)2-s + (3.96 + 6.87i)4-s + (−6.21 + 10.7i)5-s + (−18.4 − 1.73i)7-s + 3.95·8-s + (1.54 + 2.67i)10-s + (30.1 + 52.2i)11-s + 36.4·13-s + (−2.66 + 3.74i)14-s + (−31.2 + 54.1i)16-s + (−24.3 − 42.2i)17-s + (25.2 − 43.7i)19-s − 98.7·20-s + 14.9·22-s + (69.3 − 120. i)23-s + ⋯
L(s)  = 1  + (0.0438 − 0.0759i)2-s + (0.496 + 0.859i)4-s + (−0.556 + 0.963i)5-s + (−0.995 − 0.0938i)7-s + 0.174·8-s + (0.0487 + 0.0844i)10-s + (0.826 + 1.43i)11-s + 0.777·13-s + (−0.0507 + 0.0715i)14-s + (−0.488 + 0.846i)16-s + (−0.347 − 0.602i)17-s + (0.305 − 0.528i)19-s − 1.10·20-s + 0.144·22-s + (0.629 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0306 - 0.999i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.0306 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.953292 + 0.924516i\)
\(L(\frac12)\) \(\approx\) \(0.953292 + 0.924516i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (18.4 + 1.73i)T \)
good2 \( 1 + (-0.124 + 0.214i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (6.21 - 10.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.1 - 52.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 + (24.3 + 42.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-25.2 + 43.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-69.3 + 120. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 61.1T + 2.43e4T^{2} \)
31 \( 1 + (-0.584 - 1.01i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (34.7 - 60.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 308.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + (-194. + 337. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-157. - 272. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-422. - 731. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-169. + 293. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-485. - 841. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 98.4T + 3.57e5T^{2} \)
73 \( 1 + (355. + 615. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-243. + 421. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 605.T + 5.71e5T^{2} \)
89 \( 1 + (-109. + 188. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 782.T + 9.12e5T^{2} \)
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

−14.96404558581988873482359576571, −13.48448732871182961499645239432, −12.37564965654279939747258126225, −11.49266531269919576217059124303, −10.32746877392429396256218991486, −8.868064367681098216557512754833, −7.12034874804432587392200085234, −6.76086857673915214559187784528, −4.06283711919455872711754429935, −2.79506605545043200863011374633, 0.970356283978368842302507435992, 3.65401194276581920470012029282, 5.58385534520113376740187272825, 6.60096639176347016691888443397, 8.436629753000804855150440155583, 9.454205626899829107566444731969, 10.89018958680649175319040473555, 11.86838980746396149704165971462, 13.13874010953895795801116142665, 14.16150288397414218956448859363

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