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Properties

Label 2-63-7.5-c6-0-18
Degree $2$
Conductor $63$
Sign $-0.282 - 0.959i$
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  + (7.79 − 13.5i)2-s + (−89.6 − 155. i)4-s + (−22.3 − 12.9i)5-s + (−203. + 276. i)7-s − 1.79e3·8-s + (−348. + 201. i)10-s + (311. + 540. i)11-s − 3.25e3i·13-s + (2.14e3 + 4.89e3i)14-s + (−8.27e3 + 1.43e4i)16-s + (−275. + 158. i)17-s + (−5.19e3 − 3.00e3i)19-s + 4.62e3i·20-s + 9.73e3·22-s + (−21.0 + 36.3i)23-s + ⋯
L(s)  = 1  + (0.974 − 1.68i)2-s + (−1.40 − 2.42i)4-s + (−0.178 − 0.103i)5-s + (−0.592 + 0.805i)7-s − 3.50·8-s + (−0.348 + 0.201i)10-s + (0.234 + 0.405i)11-s − 1.48i·13-s + (0.783 + 1.78i)14-s + (−2.02 + 3.49i)16-s + (−0.0560 + 0.0323i)17-s + (−0.757 − 0.437i)19-s + 0.578i·20-s + 0.913·22-s + (−0.00172 + 0.00299i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(14.4934\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3),\ -0.282 - 0.959i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.784836 + 1.04959i\)
\(L(\frac12)\) \(\approx\) \(0.784836 + 1.04959i\)
\(L(4)\) 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 \( 1 + (203. - 276. i)T \)
good2 \( 1 + (-7.79 + 13.5i)T + (-32 - 55.4i)T^{2} \)
5 \( 1 + (22.3 + 12.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-311. - 540. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 3.25e3iT - 4.82e6T^{2} \)
17 \( 1 + (275. - 158. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (5.19e3 + 3.00e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (21.0 - 36.3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 2.42e4T + 5.94e8T^{2} \)
31 \( 1 + (1.75e4 - 1.01e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-8.68e3 + 1.50e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 1.00e5iT - 4.75e9T^{2} \)
43 \( 1 + 6.78e4T + 6.32e9T^{2} \)
47 \( 1 + (5.70e4 + 3.29e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-4.96e4 - 8.59e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (8.71e4 - 5.03e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (2.01e5 + 1.16e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.05e4 + 3.55e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 4.00e5T + 1.28e11T^{2} \)
73 \( 1 + (-4.79e5 + 2.76e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-5.24e3 + 9.07e3i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 7.12e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.64e5 - 9.52e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.07e6iT - 8.32e11T^{2} \)
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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

−12.59963179902933938319062710818, −12.15622148157889722789433058892, −10.80935397102842174894919653015, −9.918347103562176166476645374635, −8.715718951225095462734256406683, −6.12338325378800508799300415989, −4.91686105344580029544038256685, −3.44136491673692982782234934955, −2.24545076830203798062738217743, −0.37658757978952929228405418191, 3.54929716766722112290819308130, 4.55709613354967179436399463833, 6.23108374791281743254682165392, 6.95651584792876782983521872621, 8.186316495438557039220686194210, 9.461045370859000243427268707170, 11.54590303179982880584285908655, 12.82984824905779041032561449414, 13.73010072480633769073873235419, 14.45811450263333617083565100059

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