Properties

Label 2-63-1.1-c5-0-3
Degree $2$
Conductor $63$
Sign $1$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 31·4-s + 34·5-s − 49·7-s + 63·8-s − 34·10-s + 340·11-s + 454·13-s + 49·14-s + 929·16-s + 798·17-s + 892·19-s − 1.05e3·20-s − 340·22-s + 3.19e3·23-s − 1.96e3·25-s − 454·26-s + 1.51e3·28-s + 8.24e3·29-s − 2.49e3·31-s − 2.94e3·32-s − 798·34-s − 1.66e3·35-s + 9.79e3·37-s − 892·38-s + 2.14e3·40-s − 1.98e4·41-s + ⋯
L(s)  = 1  − 0.176·2-s − 0.968·4-s + 0.608·5-s − 0.377·7-s + 0.348·8-s − 0.107·10-s + 0.847·11-s + 0.745·13-s + 0.0668·14-s + 0.907·16-s + 0.669·17-s + 0.566·19-s − 0.589·20-s − 0.149·22-s + 1.25·23-s − 0.630·25-s − 0.131·26-s + 0.366·28-s + 1.81·29-s − 0.466·31-s − 0.508·32-s − 0.118·34-s − 0.229·35-s + 1.17·37-s − 0.100·38-s + 0.211·40-s − 1.84·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.460275632\)
\(L(\frac12)\) \(\approx\) \(1.460275632\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{2} T \)
good2 \( 1 + T + p^{5} T^{2} \)
5 \( 1 - 34 T + p^{5} T^{2} \)
11 \( 1 - 340 T + p^{5} T^{2} \)
13 \( 1 - 454 T + p^{5} T^{2} \)
17 \( 1 - 798 T + p^{5} T^{2} \)
19 \( 1 - 892 T + p^{5} T^{2} \)
23 \( 1 - 3192 T + p^{5} T^{2} \)
29 \( 1 - 8242 T + p^{5} T^{2} \)
31 \( 1 + 2496 T + p^{5} T^{2} \)
37 \( 1 - 9798 T + p^{5} T^{2} \)
41 \( 1 + 19834 T + p^{5} T^{2} \)
43 \( 1 + 17236 T + p^{5} T^{2} \)
47 \( 1 + 8928 T + p^{5} T^{2} \)
53 \( 1 + 150 T + p^{5} T^{2} \)
59 \( 1 - 42396 T + p^{5} T^{2} \)
61 \( 1 - 14758 T + p^{5} T^{2} \)
67 \( 1 + 1676 T + p^{5} T^{2} \)
71 \( 1 + 14568 T + p^{5} T^{2} \)
73 \( 1 - 78378 T + p^{5} T^{2} \)
79 \( 1 + 2272 T + p^{5} T^{2} \)
83 \( 1 - 37764 T + p^{5} T^{2} \)
89 \( 1 - 117286 T + p^{5} T^{2} \)
97 \( 1 - 10002 T + p^{5} T^{2} \)
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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

−13.82146605035704294422884079869, −13.11966601210301264113418151012, −11.78600831121606921028534122277, −10.20104993110059503481552408139, −9.371102911638747706506104033573, −8.303963228634616533879968982730, −6.55573956670720736585811620798, −5.15085999768495658742897163097, −3.52119467218360483878114964823, −1.09741881827720846859038886431, 1.09741881827720846859038886431, 3.52119467218360483878114964823, 5.15085999768495658742897163097, 6.55573956670720736585811620798, 8.303963228634616533879968982730, 9.371102911638747706506104033573, 10.20104993110059503481552408139, 11.78600831121606921028534122277, 13.11966601210301264113418151012, 13.82146605035704294422884079869

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