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Properties

Label 2-63-9.4-c3-0-7
Degree $2$
Conductor $63$
Sign $0.156 - 0.987i$
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  + (1.32 + 2.29i)2-s + (5.12 + 0.858i)3-s + (0.479 − 0.831i)4-s + (−10.0 + 17.3i)5-s + (4.82 + 12.9i)6-s + (−3.5 − 6.06i)7-s + 23.7·8-s + (25.5 + 8.79i)9-s − 53.2·10-s + (−0.517 − 0.895i)11-s + (3.17 − 3.84i)12-s + (30.1 − 52.2i)13-s + (9.28 − 16.0i)14-s + (−66.3 + 80.4i)15-s + (27.6 + 47.9i)16-s − 104.·17-s + ⋯
L(s)  = 1  + (0.469 + 0.812i)2-s + (0.986 + 0.165i)3-s + (0.0599 − 0.103i)4-s + (−0.897 + 1.55i)5-s + (0.328 + 0.878i)6-s + (−0.188 − 0.327i)7-s + 1.05·8-s + (0.945 + 0.325i)9-s − 1.68·10-s + (−0.0141 − 0.0245i)11-s + (0.0763 − 0.0925i)12-s + (0.643 − 1.11i)13-s + (0.177 − 0.307i)14-s + (−1.14 + 1.38i)15-s + (0.432 + 0.749i)16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.72471 + 1.47250i\)
\(L(\frac12)\) \(\approx\) \(1.72471 + 1.47250i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.12 - 0.858i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (-1.32 - 2.29i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (10.0 - 17.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (0.517 + 0.895i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-30.1 + 52.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 + 6.08T + 6.85e3T^{2} \)
23 \( 1 + (-105. + 182. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-24.6 - 42.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (25.8 - 44.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 198.T + 5.06e4T^{2} \)
41 \( 1 + (45.3 - 78.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-86.4 - 149. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-50.2 - 86.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 151.T + 1.48e5T^{2} \)
59 \( 1 + (174. - 303. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-73.2 - 126. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (133. - 231. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 808.T + 3.57e5T^{2} \)
73 \( 1 + 107.T + 3.89e5T^{2} \)
79 \( 1 + (253. + 439. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-590. - 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 931.T + 7.04e5T^{2} \)
97 \( 1 + (-206. - 357. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.89819148970482798531852803805, −14.02636662897602693012399875117, −13.00060711669939329714708961166, −10.88848749990197902726067155139, −10.46178253381922895293578089386, −8.439054800221900996921932319009, −7.28303526693358788723923478203, −6.53081273227506628817171271397, −4.32970944065216098800128849410, −2.91635161591254694752848750279, 1.70753783747005817348785766960, 3.65289539877669065948764830495, 4.62840180685130461180489301878, 7.24854252507768998616457329569, 8.537601036203760352965249707710, 9.251686542522690938421199291371, 11.26449835238658089424624041313, 12.13423952576429862541547777276, 13.09850850683818495608858775625, 13.64401756636062457417284179796

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