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Properties

Label 2-63-63.41-c3-0-7
Degree $2$
Conductor $63$
Sign $0.439 - 0.898i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.49i)2-s + (−1.03 + 5.09i)3-s + (0.478 − 0.828i)4-s + (−7.80 + 13.5i)5-s + (4.92 + 14.7i)6-s + (17.7 − 5.41i)7-s + 21.0i·8-s + (−24.8 − 10.5i)9-s + 46.6i·10-s + (46.4 − 26.8i)11-s + (3.72 + 3.29i)12-s + (−29.1 − 16.8i)13-s + (37.8 − 40.5i)14-s + (−60.7 − 53.7i)15-s + (35.3 + 61.2i)16-s + 43.5·17-s + ⋯
L(s)  = 1  + (0.916 − 0.529i)2-s + (−0.199 + 0.979i)3-s + (0.0598 − 0.103i)4-s + (−0.697 + 1.20i)5-s + (0.335 + 1.00i)6-s + (0.956 − 0.292i)7-s + 0.931i·8-s + (−0.920 − 0.391i)9-s + 1.47i·10-s + (1.27 − 0.735i)11-s + (0.0895 + 0.0792i)12-s + (−0.621 − 0.358i)13-s + (0.721 − 0.773i)14-s + (−1.04 − 0.925i)15-s + (0.552 + 0.957i)16-s + 0.621·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63584 + 1.02127i\)
\(L(\frac12)\) \(\approx\) \(1.63584 + 1.02127i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.03 - 5.09i)T \)
7 \( 1 + (-17.7 + 5.41i)T \)
good2 \( 1 + (-2.59 + 1.49i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (7.80 - 13.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-46.4 + 26.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (29.1 + 16.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 43.5T + 4.91e3T^{2} \)
19 \( 1 + 32.2iT - 6.85e3T^{2} \)
23 \( 1 + (-129. - 74.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (127. - 73.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (61.1 + 35.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 40.3T + 5.06e4T^{2} \)
41 \( 1 + (-179. + 310. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (253. + 439. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-228. - 395. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 213. iT - 1.48e5T^{2} \)
59 \( 1 + (159. - 276. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-303. + 175. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (289. - 501. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 787. iT - 3.57e5T^{2} \)
73 \( 1 + 146. iT - 3.89e5T^{2} \)
79 \( 1 + (193. + 334. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (98.9 + 171. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 596.T + 7.04e5T^{2} \)
97 \( 1 + (-631. + 364. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63287369013548301311251554574, −13.89980820932268893157708761535, −12.05621884353948482477930235243, −11.27112243138844935885990842914, −10.73596257232387996020323614873, −8.934230813340909872450908914738, −7.40505653988321328244783959692, −5.47387110696332497617315380449, −4.07656744056204145439987260148, −3.16327264236175769170082770271, 1.24426166420003141393534205330, 4.36587111576439119329445342138, 5.32544463383216919836580445402, 6.83616790181299068933541082298, 8.016821585790526588636735395718, 9.303305038511715777291829451221, 11.57446608070924432582146640251, 12.29046939467586298836205929484, 13.04168409029434242282763920615, 14.45579813142556630637065137202

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