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Properties

Label 2-63-63.20-c3-0-2
Degree $2$
Conductor $63$
Sign $0.759 - 0.650i$
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  + (−3.28 − 1.89i)2-s + (−4.97 + 1.49i)3-s + (3.19 + 5.53i)4-s + (−9.97 − 17.2i)5-s + (19.1 + 4.51i)6-s + (−4.13 + 18.0i)7-s + 6.12i·8-s + (22.5 − 14.9i)9-s + 75.6i·10-s + (31.1 + 17.9i)11-s + (−24.1 − 22.7i)12-s + (6.59 − 3.81i)13-s + (47.8 − 51.4i)14-s + (75.5 + 71.0i)15-s + (37.1 − 64.3i)16-s − 21.4·17-s + ⋯
L(s)  = 1  + (−1.16 − 0.670i)2-s + (−0.957 + 0.288i)3-s + (0.399 + 0.691i)4-s + (−0.892 − 1.54i)5-s + (1.30 + 0.307i)6-s + (−0.223 + 0.974i)7-s + 0.270i·8-s + (0.833 − 0.552i)9-s + 2.39i·10-s + (0.854 + 0.493i)11-s + (−0.581 − 0.546i)12-s + (0.140 − 0.0812i)13-s + (0.912 − 0.982i)14-s + (1.29 + 1.22i)15-s + (0.580 − 1.00i)16-s − 0.306·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.759 - 0.650i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.277607 + 0.102713i\)
\(L(\frac12)\) \(\approx\) \(0.277607 + 0.102713i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 - 1.49i)T \)
7 \( 1 + (4.13 - 18.0i)T \)
good2 \( 1 + (3.28 + 1.89i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (9.97 + 17.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-31.1 - 17.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.59 + 3.81i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 21.4T + 4.91e3T^{2} \)
19 \( 1 - 97.3iT - 6.85e3T^{2} \)
23 \( 1 + (35.6 - 20.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (51.8 + 29.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-61.1 + 35.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 355.T + 5.06e4T^{2} \)
41 \( 1 + (-7.30 - 12.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (48.8 - 84.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (234. - 406. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 710. iT - 1.48e5T^{2} \)
59 \( 1 + (-232. - 403. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-542. - 313. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (262. + 454. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 115. iT - 3.57e5T^{2} \)
73 \( 1 - 708. iT - 3.89e5T^{2} \)
79 \( 1 + (-128. + 222. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-200. + 347. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 977.T + 7.04e5T^{2} \)
97 \( 1 + (521. + 300. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.02509057263606624697063206714, −12.70775107925893481748593086864, −12.00218200376731379578798316550, −11.37557745520740930065333283710, −9.775065441804989201966839181203, −9.054465803696865373587479051695, −7.931659062188275214504519349747, −5.77004540322549416664140776204, −4.33799349608510719561986824031, −1.25614452422793401145485898796, 0.39235902826583682622427496629, 3.87787708027615769278658624521, 6.61994774853743903135502870937, 6.87824722729890871870083656483, 8.049175339868866441276751024629, 9.853085415538313592931824739971, 10.88152624396496199344755481949, 11.54032533460997749607562842790, 13.27662288820338038729530826696, 14.69272169633097953465291792773

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