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Properties

Label 2-63-9.4-c3-0-1
Degree $2$
Conductor $63$
Sign $0.367 - 0.930i$
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  + (−1.61 − 2.80i)2-s + (−5.00 − 1.41i)3-s + (−1.23 + 2.13i)4-s + (−4.95 + 8.57i)5-s + (4.13 + 16.2i)6-s + (−3.5 − 6.06i)7-s − 17.9·8-s + (23.0 + 14.1i)9-s + 32.0·10-s + (35.3 + 61.2i)11-s + (9.18 − 8.93i)12-s + (12.1 − 21.1i)13-s + (−11.3 + 19.6i)14-s + (36.8 − 35.8i)15-s + (38.8 + 67.2i)16-s − 112.·17-s + ⋯
L(s)  = 1  + (−0.571 − 0.990i)2-s + (−0.962 − 0.271i)3-s + (−0.154 + 0.266i)4-s + (−0.442 + 0.767i)5-s + (0.281 + 1.10i)6-s + (−0.188 − 0.327i)7-s − 0.791·8-s + (0.852 + 0.523i)9-s + 1.01·10-s + (0.969 + 1.67i)11-s + (0.220 − 0.214i)12-s + (0.260 − 0.450i)13-s + (−0.216 + 0.374i)14-s + (0.634 − 0.617i)15-s + (0.606 + 1.05i)16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.367 - 0.930i$
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: \((2,\ 63,\ (\ :3/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.219449 + 0.149303i\)
\(L(\frac12)\) \(\approx\) \(0.219449 + 0.149303i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.00 + 1.41i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (1.61 + 2.80i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (4.95 - 8.57i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.3 - 61.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-12.1 + 21.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 112.T + 4.91e3T^{2} \)
19 \( 1 + 87.5T + 6.85e3T^{2} \)
23 \( 1 + (66.6 - 115. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (98.4 + 170. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (87.5 - 151. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 16.2T + 5.06e4T^{2} \)
41 \( 1 + (-108. + 187. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-31.7 - 54.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-138. - 240. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 101.T + 1.48e5T^{2} \)
59 \( 1 + (0.975 - 1.68i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (16.8 + 29.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-171. + 297. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 908.T + 3.57e5T^{2} \)
73 \( 1 + 405.T + 3.89e5T^{2} \)
79 \( 1 + (228. + 396. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (44.7 + 77.5i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + (-150. - 261. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.89066886300905927985602689927, −13.05923303140591395558614436679, −12.06798434753582436950824818247, −11.16643644778688841764527904010, −10.44912965364941894609723819776, −9.326403235511216483143350936235, −7.30166223549317811357718477202, −6.30634931464671795982510048415, −4.13375456218000813990025663870, −1.89147752193614575608030314772, 0.23998306708814417874906059561, 4.16489032647901600298609691953, 5.94346481412699100067785208679, 6.69253426315943497612615781509, 8.548076273914447140743605824573, 9.052728476862596970275865545752, 10.98326907704087835416538398123, 11.87343199757644223208708335310, 12.98708817057034522779210853580, 14.70955717819036722775276823268

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