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Properties

Label 2-63-63.61-c2-0-0
Degree $2$
Conductor $63$
Sign $0.113 - 0.993i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 2.90i)2-s + (−2.99 + 0.230i)3-s + (−3.62 + 6.28i)4-s − 0.888i·5-s + (5.68 + 8.30i)6-s + (−3.29 + 6.17i)7-s + 10.9·8-s + (8.89 − 1.38i)9-s + (−2.58 + 1.49i)10-s − 5.50·11-s + (9.40 − 19.6i)12-s + (−12.3 + 7.11i)13-s + (23.4 − 0.792i)14-s + (0.205 + 2.65i)15-s + (−3.81 − 6.61i)16-s + (−11.2 + 6.51i)17-s + ⋯
L(s)  = 1  + (−0.838 − 1.45i)2-s + (−0.997 + 0.0769i)3-s + (−0.907 + 1.57i)4-s − 0.177i·5-s + (0.948 + 1.38i)6-s + (−0.470 + 0.882i)7-s + 1.36·8-s + (0.988 − 0.153i)9-s + (−0.258 + 0.149i)10-s − 0.500·11-s + (0.783 − 1.63i)12-s + (−0.948 + 0.547i)13-s + (1.67 − 0.0566i)14-s + (0.0136 + 0.177i)15-s + (−0.238 − 0.413i)16-s + (−0.664 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.113 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0589147 + 0.0525875i\)
\(L(\frac12)\) \(\approx\) \(0.0589147 + 0.0525875i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 - 0.230i)T \)
7 \( 1 + (3.29 - 6.17i)T \)
good2 \( 1 + (1.67 + 2.90i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + 0.888iT - 25T^{2} \)
11 \( 1 + 5.50T + 121T^{2} \)
13 \( 1 + (12.3 - 7.11i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (11.2 - 6.51i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (22.7 + 13.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + 36.0T + 529T^{2} \)
29 \( 1 + (-16.4 + 28.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-32.8 - 18.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-21.4 + 37.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (51.1 - 29.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.4 - 25.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (10.8 - 6.27i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (4.88 + 8.45i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-21.3 - 12.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.7 - 9.08i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.1 - 26.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 43.4T + 5.04e3T^{2} \)
73 \( 1 + (-23.5 + 13.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-33.7 - 58.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (8.30 + 4.79i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-54.8 - 31.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (76.9 + 44.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−15.22869579323455609070570860279, −13.11189712249638582535108164282, −12.31679368228885023366368160341, −11.59384805230777169846806615506, −10.44966221280053155087794693908, −9.619195044896670620099516959009, −8.401495436220427198813186158851, −6.39889040609558812038482677345, −4.53044342368973256125087772169, −2.32926773835169829126708220025, 0.10170637697886614009418515691, 4.78680517822910481856540784555, 6.23844915234422643786761173174, 7.05963333540820062681440306894, 8.174531797694304077597489886570, 9.954732482553038063827456599320, 10.48892777469379060994414374685, 12.28071578476478545549876913555, 13.58919679144783505914776733270, 14.91679965151554770499263063498

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