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Properties

Label 2-2e6-16.5-c11-0-0
Degree $2$
Conductor $64$
Sign $0.0876 - 0.996i$
Analytic cond. $49.1739$
Root an. cond. $7.01241$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (291. − 291. i)3-s + (−9.75e3 − 9.75e3i)5-s + 1.64e4i·7-s + 7.59e3i·9-s + (−4.77e5 − 4.77e5i)11-s + (6.65e5 − 6.65e5i)13-s − 5.68e6·15-s − 5.80e6·17-s + (−1.91e6 + 1.91e6i)19-s + (4.78e6 + 4.78e6i)21-s − 1.46e7i·23-s + 1.41e8i·25-s + (5.37e7 + 5.37e7i)27-s + (5.83e7 − 5.83e7i)29-s − 8.50e7·31-s + ⋯
L(s)  = 1  + (0.691 − 0.691i)3-s + (−1.39 − 1.39i)5-s + 0.369i·7-s + 0.0428i·9-s + (−0.894 − 0.894i)11-s + (0.497 − 0.497i)13-s − 1.93·15-s − 0.991·17-s + (−0.177 + 0.177i)19-s + (0.255 + 0.255i)21-s − 0.475i·23-s + 2.89i·25-s + (0.721 + 0.721i)27-s + (0.528 − 0.528i)29-s − 0.533·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0876 - 0.996i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.0876 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.0876 - 0.996i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ 0.0876 - 0.996i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0723807 + 0.0662904i\)
\(L(\frac12)\) \(\approx\) \(0.0723807 + 0.0662904i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-291. + 291. i)T - 1.77e5iT^{2} \)
5 \( 1 + (9.75e3 + 9.75e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 1.64e4iT - 1.97e9T^{2} \)
11 \( 1 + (4.77e5 + 4.77e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-6.65e5 + 6.65e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 5.80e6T + 3.42e13T^{2} \)
19 \( 1 + (1.91e6 - 1.91e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 1.46e7iT - 9.52e14T^{2} \)
29 \( 1 + (-5.83e7 + 5.83e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 8.50e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.52e7 - 2.52e7i)T + 1.77e17iT^{2} \)
41 \( 1 - 9.24e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.00e9 + 1.00e9i)T + 9.29e17iT^{2} \)
47 \( 1 - 2.86e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.34e9 - 2.34e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-5.55e8 - 5.55e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (2.31e9 - 2.31e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (1.02e10 - 1.02e10i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.39e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.22e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.84e10T + 7.47e20T^{2} \)
83 \( 1 + (-4.10e9 + 4.10e9i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.47e10iT - 2.77e21T^{2} \)
97 \( 1 + 9.56e10T + 7.15e21T^{2} \)
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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

−12.97243181279736358741111432778, −11.98427795198198805490106054550, −10.82556665597706766586660540548, −8.658777366166169111064677152805, −8.452621366930467679576899432581, −7.40918002959143519676010306345, −5.48283733705231145550523433607, −4.17486113109063230944205476244, −2.71657916115434887657511562643, −1.08965341756077913841421945520, 0.02677782682657898701635980221, 2.49922122935165251158003840258, 3.60503931798766563109455362967, 4.41465490364335205553145138915, 6.72159088680219895867070007312, 7.58118091701896029219997103224, 8.815176410868145153325021881251, 10.26320212954317228430002590842, 10.97802893460256433026679100401, 12.14909073278667768900516231236

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