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Properties

Label 2-63-9.5-c2-0-7
Degree $2$
Conductor $63$
Sign $0.859 + 0.511i$
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  + (2.98 − 1.72i)2-s + (−1.04 + 2.81i)3-s + (3.95 − 6.84i)4-s + (0.855 + 0.493i)5-s + (1.72 + 10.2i)6-s + (−1.32 − 2.29i)7-s − 13.4i·8-s + (−6.81 − 5.87i)9-s + 3.40·10-s + (−6.31 + 3.64i)11-s + (15.1 + 18.2i)12-s + (−10.9 + 19.0i)13-s + (−7.90 − 4.56i)14-s + (−2.28 + 1.88i)15-s + (−7.44 − 12.8i)16-s − 24.8i·17-s + ⋯
L(s)  = 1  + (1.49 − 0.862i)2-s + (−0.348 + 0.937i)3-s + (0.988 − 1.71i)4-s + (0.171 + 0.0987i)5-s + (0.287 + 1.70i)6-s + (−0.188 − 0.327i)7-s − 1.68i·8-s + (−0.757 − 0.653i)9-s + 0.340·10-s + (−0.574 + 0.331i)11-s + (1.26 + 1.52i)12-s + (−0.845 + 1.46i)13-s + (−0.564 − 0.326i)14-s + (−0.152 + 0.125i)15-s + (−0.465 − 0.805i)16-s − 1.46i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.07664 - 0.571809i\)
\(L(\frac12)\) \(\approx\) \(2.07664 - 0.571809i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.04 - 2.81i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good2 \( 1 + (-2.98 + 1.72i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-0.855 - 0.493i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.31 - 3.64i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.9 - 19.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 24.8iT - 289T^{2} \)
19 \( 1 - 7.54T + 361T^{2} \)
23 \( 1 + (-22.0 - 12.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.1 + 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (12.5 - 21.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 17.5T + 1.36e3T^{2} \)
41 \( 1 + (32.7 + 18.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-24.2 - 42.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-26.9 + 15.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 12.4iT - 2.80e3T^{2} \)
59 \( 1 + (2.66 + 1.54i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.2 + 26.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (63.4 - 109. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 72.0iT - 5.04e3T^{2} \)
73 \( 1 - 26.2T + 5.32e3T^{2} \)
79 \( 1 + (3.92 + 6.80i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (59.2 - 34.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 68.5iT - 7.92e3T^{2} \)
97 \( 1 + (27.0 + 46.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.28430662719302269449219665778, −13.70539256793005596419639100772, −12.17549498734678641924351393472, −11.53384191292711599583755336170, −10.36163439735800327763401282893, −9.455003275617911617722580417981, −6.80527791382433982598962602990, −5.21933457744907333890834319123, −4.37493608003450422653800357744, −2.78923215928559826070170519945, 2.96596813014366630881765909241, 5.17571048341961993370287397158, 5.94626059648626202318353914741, 7.23423836276349428923213802119, 8.271480780233390192896168211152, 10.65249970546096417413153949302, 12.21888318698703474732538698949, 12.79630980624521234135544229366, 13.54837180973759574526884900324, 14.75683520526802508946282631228

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