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Properties

Label 2-63-21.17-c5-0-4
Degree $2$
Conductor $63$
Sign $-0.424 - 0.905i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.424 - 0.905i$
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: \((2,\ 63,\ (\ :5/2),\ -0.424 - 0.905i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.638894 + 1.00542i\)
\(L(\frac12)\) \(\approx\) \(0.638894 + 1.00542i\)
\(L(\frac{7}{2})\) 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

−14.55899631299623423333683836286, −13.54577779067718038160907526130, −11.83371338351284124512461083551, −10.39624349017668198261376503416, −9.817772325676972052430045184950, −8.351979957749954857646532251931, −7.07912334358530216076260134289, −6.37313383655600473768118874145, −3.94571007087733974639142076986, −1.62148501976673865596538024552, 0.873741395219544071405617642311, 1.99512523820441675280306027576, 4.71356928563785841247571897622, 6.07106321294325937279261228683, 8.455269719480743015536837793954, 8.905652426069110836461787692580, 9.806496014316967118192014149598, 11.44109189327703412456824821707, 12.01204349724531389860576135393, 13.65291917830880584077261434691

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