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Properties

Label 2-63-7.4-c5-0-5
Degree $2$
Conductor $63$
Sign $-0.693 - 0.720i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.19 + 5.53i)2-s + (−4.41 + 7.64i)4-s + (19.3 + 33.5i)5-s + (−87.5 + 95.6i)7-s + 148.·8-s + (−123. + 214. i)10-s + (−288. + 499. i)11-s + 391.·13-s + (−808. − 178. i)14-s + (614. + 1.06e3i)16-s + (−664. + 1.15e3i)17-s + (−471. − 816. i)19-s − 341.·20-s − 3.68e3·22-s + (−816. − 1.41e3i)23-s + ⋯
L(s)  = 1  + (0.564 + 0.978i)2-s + (−0.137 + 0.238i)4-s + (0.346 + 0.599i)5-s + (−0.674 + 0.737i)7-s + 0.817·8-s + (−0.391 + 0.677i)10-s + (−0.718 + 1.24i)11-s + 0.642·13-s + (−1.10 − 0.243i)14-s + (0.599 + 1.03i)16-s + (−0.557 + 0.966i)17-s + (−0.299 − 0.518i)19-s − 0.191·20-s − 1.62·22-s + (−0.321 − 0.557i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.693 - 0.720i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.693 - 0.720i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.883186 + 2.07660i\)
\(L(\frac12)\) \(\approx\) \(0.883186 + 2.07660i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (87.5 - 95.6i)T \)
good2 \( 1 + (-3.19 - 5.53i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-19.3 - 33.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (288. - 499. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 391.T + 3.71e5T^{2} \)
17 \( 1 + (664. - 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (471. + 816. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (816. + 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.46e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.95e3 + 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.31e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + (3.40e3 + 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.00e3 - 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.05e4 + 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.99e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.78e4 + 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.15e4 - 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 4.55e4T + 3.93e9T^{2} \)
89 \( 1 + (-7.84e3 - 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.12e3T + 8.58e9T^{2} \)
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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

−14.68151783989654694458387746094, −13.39574908955859247085733735743, −12.59956471139530015840316235422, −10.84297047278288871949512798150, −9.830360321556640579026806338965, −8.180035090900348004371191503094, −6.71873400840573116480909909272, −6.02826249314955159326179488984, −4.50870537155258379275123571692, −2.36558100513748689796042375319, 0.913659814833482489122263478614, 2.86225991646260963046018128783, 4.17474349886671882628330907561, 5.77488693490119691958478933538, 7.55263250614720839577557188872, 9.089910037857500450007260090837, 10.48094318499489890604593045445, 11.25998872055264165015616786521, 12.63170020607479222791478660120, 13.38044492524423633010194496177

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