Properties

Label 2-63-7.3-c6-0-7
Degree $2$
Conductor $63$
Sign $-0.450 - 0.892i$
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  + (4.12 + 7.13i)2-s + (−1.97 + 3.41i)4-s + (−68.5 + 39.5i)5-s + (337. + 62.3i)7-s + 495.·8-s + (−565. − 326. i)10-s + (−854. + 1.47e3i)11-s + 3.12e3i·13-s + (945. + 2.66e3i)14-s + (2.16e3 + 3.75e3i)16-s + (−3.52e3 − 2.03e3i)17-s + (5.08e3 − 2.93e3i)19-s − 312. i·20-s − 1.40e4·22-s + (6.66e3 + 1.15e4i)23-s + ⋯
L(s)  = 1  + (0.515 + 0.892i)2-s + (−0.0307 + 0.0533i)4-s + (−0.548 + 0.316i)5-s + (0.983 + 0.181i)7-s + 0.966·8-s + (−0.565 − 0.326i)10-s + (−0.641 + 1.11i)11-s + 1.42i·13-s + (0.344 + 0.971i)14-s + (0.528 + 0.916i)16-s + (−0.718 − 0.414i)17-s + (0.741 − 0.428i)19-s − 0.0390i·20-s − 1.32·22-s + (0.547 + 0.948i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.450 - 0.892i$
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: \((2,\ 63,\ (\ :3),\ -0.450 - 0.892i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.25739 + 2.04346i\)
\(L(\frac12)\) \(\approx\) \(1.25739 + 2.04346i\)
\(L(4)\) 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 \( 1 + (-337. - 62.3i)T \)
good2 \( 1 + (-4.12 - 7.13i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (68.5 - 39.5i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (854. - 1.47e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 3.12e3iT - 4.82e6T^{2} \)
17 \( 1 + (3.52e3 + 2.03e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-5.08e3 + 2.93e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-6.66e3 - 1.15e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 6.51e3T + 5.94e8T^{2} \)
31 \( 1 + (1.03e4 + 5.99e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-2.32e3 - 4.01e3i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 1.93e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.16e4T + 6.32e9T^{2} \)
47 \( 1 + (-5.58e4 + 3.22e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-7.47e4 + 1.29e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (5.28e4 + 3.05e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (8.54e4 - 4.93e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.55e5 + 2.70e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.01e5T + 1.28e11T^{2} \)
73 \( 1 + (5.82e5 + 3.36e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-1.60e5 - 2.77e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 8.32e5iT - 3.26e11T^{2} \)
89 \( 1 + (3.28e5 - 1.89e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.05e6iT - 8.32e11T^{2} \)
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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

−14.31914489464171954451341983585, −13.36662799872645727712363803648, −11.75280217292382499381478311554, −10.93443092049972860007847069161, −9.308192927534523110889590369648, −7.61145314414445431969897914290, −7.02372669191929613813184125700, −5.30071053663033170991132751364, −4.32304581683783431886266695593, −1.89681856783452089255377299312, 0.875719877596343238005355022533, 2.73278583269853894420716356961, 4.11754344239589234442598426057, 5.42821589213719395640699088867, 7.66510591067898884545951150869, 8.415801245015061026756882552066, 10.54909961466582104083823414389, 11.10221811133055863811612158439, 12.24503180285734244880831480930, 13.13926873813963073428212279512

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