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Properties

Label 2-63-21.17-c5-0-12
Degree $2$
Conductor $63$
Sign $-0.711 + 0.702i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.81 − 3.35i)2-s + (6.53 − 11.3i)4-s + (−43.0 − 74.5i)5-s + (95.4 − 87.7i)7-s + 127. i·8-s + (−500. − 288. i)10-s + (−668. − 386. i)11-s − 273. i·13-s + (260. − 830. i)14-s + (635. + 1.10e3i)16-s + (563. − 975. i)17-s + (−303. + 175. i)19-s − 1.12e3·20-s − 5.18e3·22-s + (2.84e3 − 1.64e3i)23-s + ⋯
L(s)  = 1  + (1.02 − 0.593i)2-s + (0.204 − 0.353i)4-s + (−0.769 − 1.33i)5-s + (0.736 − 0.676i)7-s + 0.702i·8-s + (−1.58 − 0.913i)10-s + (−1.66 − 0.962i)11-s − 0.449i·13-s + (0.354 − 1.13i)14-s + (0.620 + 1.07i)16-s + (0.472 − 0.818i)17-s + (−0.192 + 0.111i)19-s − 0.628·20-s − 2.28·22-s + (1.11 − 0.646i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.711 + 0.702i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.824711 - 2.01067i\)
\(L(\frac12)\) \(\approx\) \(0.824711 - 2.01067i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-95.4 + 87.7i)T \)
good2 \( 1 + (-5.81 + 3.35i)T + (16 - 27.7i)T^{2} \)
5 \( 1 + (43.0 + 74.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (668. + 386. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 273. iT - 3.71e5T^{2} \)
17 \( 1 + (-563. + 975. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (303. - 175. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.84e3 + 1.64e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.05e3iT - 2.05e7T^{2} \)
31 \( 1 + (-7.39e3 - 4.26e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.12e3 - 1.94e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 9.76e3T + 1.15e8T^{2} \)
43 \( 1 + 5.73e3T + 1.47e8T^{2} \)
47 \( 1 + (-6.59e3 - 1.14e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.89e4 + 1.09e4i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-7.84e3 + 1.35e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-3.85e4 + 2.22e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.45e3 - 1.29e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.71e4iT - 1.80e9T^{2} \)
73 \( 1 + (3.69e4 + 2.13e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.85e3 - 6.67e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 2.63e3T + 3.93e9T^{2} \)
89 \( 1 + (5.21e4 + 9.02e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.26e3iT - 8.58e9T^{2} \)
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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

−13.26854740953357750199577300575, −12.63565072653250722212552777357, −11.51629364434390709415783474388, −10.60972763211120006228258372962, −8.499710019844616389710730128928, −7.85484562968042435525073688674, −5.23842880977922472862105071695, −4.64744982225413241221329712984, −3.06951079843149425044587905343, −0.75285012463895074735068480989, 2.70734026087317359954806510307, 4.34872044222759647798334079929, 5.62475291438490704420910450059, 7.04610261941031574634736747732, 7.951959878206286769145914140792, 10.02720959305986301820927435169, 11.15665080503521469477674817047, 12.34128344214771757845975852831, 13.47700929965013416696147529026, 14.79509318740465347844864606741

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