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Properties

Label 2-63-63.38-c3-0-5
Degree $2$
Conductor $63$
Sign $-0.996 + 0.0796i$
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  + 4.10i·2-s + (1.00 + 5.09i)3-s − 8.86·4-s + (−3.80 + 6.59i)5-s + (−20.9 + 4.14i)6-s + (13.1 − 13.0i)7-s − 3.56i·8-s + (−24.9 + 10.2i)9-s + (−27.0 − 15.6i)10-s + (6.20 − 3.58i)11-s + (−8.94 − 45.2i)12-s + (59.6 − 34.4i)13-s + (53.7 + 53.8i)14-s + (−37.4 − 12.7i)15-s − 56.2·16-s + (−11.4 + 19.8i)17-s + ⋯
L(s)  = 1  + 1.45i·2-s + (0.194 + 0.980i)3-s − 1.10·4-s + (−0.340 + 0.589i)5-s + (−1.42 + 0.281i)6-s + (0.708 − 0.706i)7-s − 0.157i·8-s + (−0.924 + 0.380i)9-s + (−0.856 − 0.494i)10-s + (0.170 − 0.0982i)11-s + (−0.215 − 1.08i)12-s + (1.27 − 0.734i)13-s + (1.02 + 1.02i)14-s + (−0.644 − 0.219i)15-s − 0.879·16-s + (−0.163 + 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0796i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.996 + 0.0796i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.996 + 0.0796i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0570966 - 1.43210i\)
\(L(\frac12)\) \(\approx\) \(0.0570966 - 1.43210i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.00 - 5.09i)T \)
7 \( 1 + (-13.1 + 13.0i)T \)
good2 \( 1 - 4.10iT - 8T^{2} \)
5 \( 1 + (3.80 - 6.59i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-6.20 + 3.58i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-59.6 + 34.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (11.4 - 19.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (86.1 - 49.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-106. - 61.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-143. - 82.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 13.7iT - 2.97e4T^{2} \)
37 \( 1 + (-126. - 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (159. + 276. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-156. + 271. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 70.7T + 1.03e5T^{2} \)
53 \( 1 + (254. + 146. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 461.T + 2.05e5T^{2} \)
61 \( 1 + 662. iT - 2.26e5T^{2} \)
67 \( 1 - 718.T + 3.00e5T^{2} \)
71 \( 1 - 92.9iT - 3.57e5T^{2} \)
73 \( 1 + (882. + 509. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 796.T + 4.93e5T^{2} \)
83 \( 1 + (10.7 - 18.5i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-339. - 588. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-984. - 568. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.12212892829135762872446011040, −14.45183473073712126459794876835, −13.45619037593274543834521713982, −11.21906058371326822072751947643, −10.53681365593636409980852843773, −8.739017283049288612854176623702, −7.954278209378560019926058151113, −6.55992268054752472441070638618, −5.15832234340614558990178669977, −3.72369398657796481816904132304, 1.10159363271243750513945522875, 2.52554696663640349254092637329, 4.44594730615360835675958715975, 6.51167144013610875228700635116, 8.407150561145269129441104809830, 9.066929779220717565083472823864, 11.01327510441479444545651150483, 11.66823439231258997701724876826, 12.59875563501423146900382000459, 13.37077974333481548592746916013

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