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Properties

Label 2-63-21.17-c7-0-10
Degree $2$
Conductor $63$
Sign $0.982 + 0.187i$
Analytic cond. $19.6802$
Root an. cond. $4.43624$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−18.8 + 10.8i)2-s + (172. − 297. i)4-s + (38.2 + 66.1i)5-s + (884. + 201. i)7-s + 4.69e3i·8-s + (−1.43e3 − 830. i)10-s + (−338. − 195. i)11-s − 9.29e3i·13-s + (−1.88e4 + 5.82e3i)14-s + (−2.89e4 − 5.01e4i)16-s + (1.01e4 − 1.75e4i)17-s + (−4.12e4 + 2.38e4i)19-s + 2.62e4·20-s + 8.48e3·22-s + (−5.30e4 + 3.06e4i)23-s + ⋯
L(s)  = 1  + (−1.66 + 0.960i)2-s + (1.34 − 2.32i)4-s + (0.136 + 0.236i)5-s + (0.975 + 0.221i)7-s + 3.24i·8-s + (−0.454 − 0.262i)10-s + (−0.0766 − 0.0442i)11-s − 1.17i·13-s + (−1.83 + 0.567i)14-s + (−1.76 − 3.06i)16-s + (0.501 − 0.868i)17-s + (−1.37 + 0.796i)19-s + 0.734·20-s + 0.169·22-s + (−0.908 + 0.524i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(19.6802\)
Root analytic conductor: \(4.43624\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :7/2),\ 0.982 + 0.187i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.780956 - 0.0740053i\)
\(L(\frac12)\) \(\approx\) \(0.780956 - 0.0740053i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-884. - 201. i)T \)
good2 \( 1 + (18.8 - 10.8i)T + (64 - 110. i)T^{2} \)
5 \( 1 + (-38.2 - 66.1i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (338. + 195. i)T + (9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + 9.29e3iT - 6.27e7T^{2} \)
17 \( 1 + (-1.01e4 + 1.75e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (4.12e4 - 2.38e4i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (5.30e4 - 3.06e4i)T + (1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + 1.36e5iT - 1.72e10T^{2} \)
31 \( 1 + (-2.25e5 - 1.30e5i)T + (1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (4.99e4 + 8.65e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 5.63e4T + 1.94e11T^{2} \)
43 \( 1 - 1.68e4T + 2.71e11T^{2} \)
47 \( 1 + (6.38e5 + 1.10e6i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-8.22e5 - 4.74e5i)T + (5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-6.91e5 + 1.19e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-2.50e6 + 1.44e6i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.28e6 + 2.21e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 4.17e6iT - 9.09e12T^{2} \)
73 \( 1 + (7.16e5 + 4.13e5i)T + (5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-2.32e6 - 4.02e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 6.11e6T + 2.71e13T^{2} \)
89 \( 1 + (1.09e6 + 1.90e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 3.26e6iT - 8.07e13T^{2} \)
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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

−14.04418374632796369761416427115, −11.90330383202100227708819563629, −10.62807131962521145010290247271, −9.923396095348971651729065090184, −8.395537435699786318367622575628, −7.902832545589091251747881300777, −6.45388415086579962974953986896, −5.27834488388230950513279666913, −2.12298866478104845591365591478, −0.56559037413349577110976035761, 1.16863305820783200958322172542, 2.25734392161315708666424338642, 4.19991223373421822858691644502, 6.78714460448712760208400692358, 8.140431447549642904750027028544, 8.862838104758773746392277133336, 10.12797825306932529550471362059, 11.05403625976599338762787417425, 11.91944968819872576665433149626, 13.07230995752348330670605316839

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