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Properties

Label 2-63-7.3-c6-0-15
Degree $2$
Conductor $63$
Sign $-0.00656 + 0.999i$
Analytic cond. $14.4934$
Root an. cond. $3.80702$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.65 + 8.06i)2-s + (−11.3 + 19.6i)4-s + (−151. + 87.3i)5-s + (−271. − 209. i)7-s + 384.·8-s + (−1.40e3 − 813. i)10-s + (−92.6 + 160. i)11-s − 3.98e3i·13-s + (424. − 3.16e3i)14-s + (2.51e3 + 4.35e3i)16-s + (−6.10e3 − 3.52e3i)17-s + (−4.06e3 + 2.34e3i)19-s − 3.95e3i·20-s − 1.72e3·22-s + (−3.32e3 − 5.75e3i)23-s + ⋯
L(s)  = 1  + (0.581 + 1.00i)2-s + (−0.176 + 0.306i)4-s + (−1.21 + 0.699i)5-s + (−0.791 − 0.610i)7-s + 0.751·8-s + (−1.40 − 0.813i)10-s + (−0.0696 + 0.120i)11-s − 1.81i·13-s + (0.154 − 1.15i)14-s + (0.614 + 1.06i)16-s + (−1.24 − 0.716i)17-s + (−0.592 + 0.342i)19-s − 0.494i·20-s − 0.162·22-s + (−0.273 − 0.472i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00656 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.00656 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.00656 + 0.999i$
Analytic conductor: \(14.4934\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3),\ -0.00656 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.308061 - 0.310091i\)
\(L(\frac12)\) \(\approx\) \(0.308061 - 0.310091i\)
\(L(4)\) 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 \( 1 + (271. + 209. i)T \)
good2 \( 1 + (-4.65 - 8.06i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (151. - 87.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (92.6 - 160. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 3.98e3iT - 4.82e6T^{2} \)
17 \( 1 + (6.10e3 + 3.52e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (4.06e3 - 2.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (3.32e3 + 5.75e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 1.93e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.07e4 - 1.19e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.97e4 - 3.42e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.31e4T + 6.32e9T^{2} \)
47 \( 1 + (1.29e5 - 7.47e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (5.09e4 - 8.81e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (1.31e5 + 7.61e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.45e5 - 8.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.05e4 - 5.29e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.85e5T + 1.28e11T^{2} \)
73 \( 1 + (7.73e3 + 4.46e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.21e5 + 3.84e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 5.59e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.79e5 + 1.03e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.23e6iT - 8.32e11T^{2} \)
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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

−13.60936646432183720759979114306, −12.68087709013573049428574219136, −11.08023407059686368305118485609, −10.21970198621596417351823956788, −8.114192458111398982324898424830, −7.22704149982349765835168484499, −6.25921621834533528037439602933, −4.57764890793342583966692050424, −3.24386794146497342135078290495, −0.13976195746827222360761963914, 2.01703089857938073765570228690, 3.72215772702598021602376049643, 4.59063637629914631976030774994, 6.67303798147190255354117603428, 8.286565028470425194990027555479, 9.450278681346612022817233533255, 11.17755101901446608912026899486, 11.78222613499451380592554488396, 12.70648379453922419046271941273, 13.51142579789262821524061148750

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