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Properties

Label 2-2e6-16.13-c21-0-10
Degree $2$
Conductor $64$
Sign $0.762 - 0.646i$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36e5 − 1.36e5i)3-s + (−8.67e6 + 8.67e6i)5-s + 5.12e8i·7-s + 2.66e10i·9-s + (1.13e10 − 1.13e10i)11-s + (2.78e11 + 2.78e11i)13-s + 2.36e12·15-s + 1.25e13·17-s + (−1.60e13 − 1.60e13i)19-s + (6.97e13 − 6.97e13i)21-s − 2.73e14i·23-s + 3.26e14i·25-s + (2.20e15 − 2.20e15i)27-s + (−9.06e14 − 9.06e14i)29-s + 4.28e15·31-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)3-s + (−0.397 + 0.397i)5-s + 0.685i·7-s + 2.54i·9-s + (0.131 − 0.131i)11-s + (0.561 + 0.561i)13-s + 1.05·15-s + 1.50·17-s + (−0.599 − 0.599i)19-s + (0.912 − 0.912i)21-s − 1.37i·23-s + 0.684i·25-s + (2.05 − 2.05i)27-s + (−0.400 − 0.400i)29-s + 0.938·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9619492007\)
\(L(\frac12)\) \(\approx\) \(0.9619492007\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (1.36e5 + 1.36e5i)T + 1.04e10iT^{2} \)
5 \( 1 + (8.67e6 - 8.67e6i)T - 4.76e14iT^{2} \)
7 \( 1 - 5.12e8iT - 5.58e17T^{2} \)
11 \( 1 + (-1.13e10 + 1.13e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-2.78e11 - 2.78e11i)T + 2.47e23iT^{2} \)
17 \( 1 - 1.25e13T + 6.90e25T^{2} \)
19 \( 1 + (1.60e13 + 1.60e13i)T + 7.14e26iT^{2} \)
23 \( 1 + 2.73e14iT - 3.94e28T^{2} \)
29 \( 1 + (9.06e14 + 9.06e14i)T + 5.13e30iT^{2} \)
31 \( 1 - 4.28e15T + 2.08e31T^{2} \)
37 \( 1 + (2.27e16 - 2.27e16i)T - 8.55e32iT^{2} \)
41 \( 1 - 1.22e17iT - 7.38e33T^{2} \)
43 \( 1 + (1.04e17 - 1.04e17i)T - 2.00e34iT^{2} \)
47 \( 1 - 5.24e17T + 1.30e35T^{2} \)
53 \( 1 + (-7.87e17 + 7.87e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (3.70e18 - 3.70e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (2.64e18 + 2.64e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (5.83e18 + 5.83e18i)T + 2.22e38iT^{2} \)
71 \( 1 - 2.18e19iT - 7.52e38T^{2} \)
73 \( 1 + 5.08e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.22e20T + 7.08e39T^{2} \)
83 \( 1 + (4.17e18 + 4.17e18i)T + 1.99e40iT^{2} \)
89 \( 1 + 1.65e19iT - 8.65e40T^{2} \)
97 \( 1 - 3.10e20T + 5.27e41T^{2} \)
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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

−11.40662209069433237633666287862, −10.35196203326395431283036700841, −8.555397866237176010411093126628, −7.50327848172601122458453230826, −6.51323674919839274772555276274, −5.84163777247163562615390139755, −4.65421799344084077861876245261, −2.88136061873554116670050539124, −1.66944800952410983563717468454, −0.74556107336235377284106083837, 0.34733658933732992185430475903, 1.15837784831174894231530140198, 3.54765096751038087957711268885, 4.06387366004921681166903915497, 5.26072014528474121654896452676, 5.95506802666098264143064641228, 7.41843065148507674670681416105, 8.876926318994387387104371952499, 10.16011023322840085408448522071, 10.56907650705930759251909791885

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