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Properties

Label 2-63-63.38-c3-0-1
Degree $2$
Conductor $63$
Sign $0.529 - 0.848i$
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  − 5.23i·2-s + (−2.33 + 4.64i)3-s − 19.4·4-s + (−6.10 + 10.5i)5-s + (24.3 + 12.2i)6-s + (18.4 + 1.88i)7-s + 59.9i·8-s + (−16.0 − 21.6i)9-s + (55.3 + 31.9i)10-s + (−55.0 + 31.8i)11-s + (45.3 − 90.2i)12-s + (−13.7 + 7.96i)13-s + (9.89 − 96.5i)14-s + (−34.8 − 53.0i)15-s + 158.·16-s + (−17.9 + 31.0i)17-s + ⋯
L(s)  = 1  − 1.85i·2-s + (−0.449 + 0.893i)3-s − 2.43·4-s + (−0.546 + 0.945i)5-s + (1.65 + 0.832i)6-s + (0.994 + 0.101i)7-s + 2.64i·8-s + (−0.596 − 0.802i)9-s + (1.75 + 1.01i)10-s + (−1.50 + 0.871i)11-s + (1.09 − 2.17i)12-s + (−0.294 + 0.169i)13-s + (0.188 − 1.84i)14-s + (−0.599 − 0.912i)15-s + 2.47·16-s + (−0.255 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.529 - 0.848i$
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: \((2,\ 63,\ (\ :3/2),\ 0.529 - 0.848i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.428672 + 0.237652i\)
\(L(\frac12)\) \(\approx\) \(0.428672 + 0.237652i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.33 - 4.64i)T \)
7 \( 1 + (-18.4 - 1.88i)T \)
good2 \( 1 + 5.23iT - 8T^{2} \)
5 \( 1 + (6.10 - 10.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (55.0 - 31.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (13.7 - 7.96i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (17.9 - 31.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13.8 - 7.97i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-39.5 - 22.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (206. + 119. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 135. iT - 2.97e4T^{2} \)
37 \( 1 + (-18.0 - 31.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-84.5 - 146. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (37.8 - 65.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 6.93T + 1.03e5T^{2} \)
53 \( 1 + (-33.2 - 19.2i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 124.T + 2.05e5T^{2} \)
61 \( 1 - 514. iT - 2.26e5T^{2} \)
67 \( 1 - 335.T + 3.00e5T^{2} \)
71 \( 1 + 716. iT - 3.57e5T^{2} \)
73 \( 1 + (-725. - 418. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 409.T + 4.93e5T^{2} \)
83 \( 1 + (135. - 234. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-717. - 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (433. + 250. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.68600229307345441558775864021, −13.15477346163125971630941254909, −11.91542801410564153054170350570, −11.07520505335310105479206965044, −10.54719674460175427797783562408, −9.483664483543364801747945401343, −7.890442361190419926698242061828, −5.14620710956755816175659906467, −4.02352297106829087365808915981, −2.47810825092906890352358883854, 0.35456991571847931837916822913, 4.88282387571897591755444259788, 5.52686012223809700216879040740, 7.20724090932399392633932693423, 8.045224787111196355683108525285, 8.712150075777611935616458747907, 10.97301547342553953777945018529, 12.56330644262582138205235142119, 13.38492830826442051668247563067, 14.36778858939291916799241284371

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