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Properties

Label 2-2e6-16.13-c21-0-16
Degree $2$
Conductor $64$
Sign $0.561 - 0.827i$
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  + (−8.01e4 − 8.01e4i)3-s + (−2.92e7 + 2.92e7i)5-s + 9.19e8i·7-s + 2.39e9i·9-s + (−8.07e10 + 8.07e10i)11-s + (2.85e11 + 2.85e11i)13-s + 4.68e12·15-s + 2.16e12·17-s + (−1.42e13 − 1.42e13i)19-s + (7.36e13 − 7.36e13i)21-s + 2.48e13i·23-s − 1.22e15i·25-s + (−6.46e14 + 6.46e14i)27-s + (1.64e15 + 1.64e15i)29-s + 6.36e15·31-s + ⋯
L(s)  = 1  + (−0.783 − 0.783i)3-s + (−1.33 + 1.33i)5-s + 1.22i·7-s + 0.228i·9-s + (−0.939 + 0.939i)11-s + (0.573 + 0.573i)13-s + 2.09·15-s + 0.260·17-s + (−0.532 − 0.532i)19-s + (0.963 − 0.963i)21-s + 0.124i·23-s − 2.57i·25-s + (−0.604 + 0.604i)27-s + (0.726 + 0.726i)29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.561 - 0.827i$
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: \((2,\ 64,\ (\ :21/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9445924687\)
\(L(\frac12)\) \(\approx\) \(0.9445924687\)
\(L(\frac{23}{2})\) 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 \( 1 + (8.01e4 + 8.01e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (2.92e7 - 2.92e7i)T - 4.76e14iT^{2} \)
7 \( 1 - 9.19e8iT - 5.58e17T^{2} \)
11 \( 1 + (8.07e10 - 8.07e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-2.85e11 - 2.85e11i)T + 2.47e23iT^{2} \)
17 \( 1 - 2.16e12T + 6.90e25T^{2} \)
19 \( 1 + (1.42e13 + 1.42e13i)T + 7.14e26iT^{2} \)
23 \( 1 - 2.48e13iT - 3.94e28T^{2} \)
29 \( 1 + (-1.64e15 - 1.64e15i)T + 5.13e30iT^{2} \)
31 \( 1 - 6.36e15T + 2.08e31T^{2} \)
37 \( 1 + (-1.90e16 + 1.90e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 1.04e17iT - 7.38e33T^{2} \)
43 \( 1 + (-1.09e17 + 1.09e17i)T - 2.00e34iT^{2} \)
47 \( 1 + 1.74e17T + 1.30e35T^{2} \)
53 \( 1 + (3.51e17 - 3.51e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (2.18e18 - 2.18e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (-2.13e18 - 2.13e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (-5.18e18 - 5.18e18i)T + 2.22e38iT^{2} \)
71 \( 1 + 2.16e19iT - 7.52e38T^{2} \)
73 \( 1 - 6.31e19iT - 1.34e39T^{2} \)
79 \( 1 + 5.28e19T + 7.08e39T^{2} \)
83 \( 1 + (1.28e20 + 1.28e20i)T + 1.99e40iT^{2} \)
89 \( 1 + 3.76e20iT - 8.65e40T^{2} \)
97 \( 1 - 1.20e21T + 5.27e41T^{2} \)
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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

−11.35956998739600957094926465932, −10.37017915725662486414187371471, −8.677976140770035777085672992768, −7.49106674997451614283929036480, −6.78929389678567591846695395851, −5.83211112909496970351358858363, −4.39920572278596107507116067373, −3.01278292211956629813649866490, −2.12449072147415964909080498497, −0.50958447148106155888702909284, 0.49953175144644317276048417277, 0.951209425809156916183654385904, 3.30144002615055006498583415367, 4.31951480937981253572003541124, 4.84125286446398225983085236975, 6.05929900100349914936592778358, 7.945910799864412046999024259899, 8.194545581164246788856952932266, 9.940909385156039230660974655301, 10.86756341582069685571784153770

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