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Properties

Label 2-2e6-16.13-c21-0-14
Degree $2$
Conductor $64$
Sign $-0.812 + 0.583i$
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  + (1.27e5 + 1.27e5i)3-s + (6.46e6 − 6.46e6i)5-s + 1.09e9i·7-s + 2.21e10i·9-s + (−9.04e10 + 9.04e10i)11-s + (6.65e10 + 6.65e10i)13-s + 1.65e12·15-s − 1.23e13·17-s + (2.96e13 + 2.96e13i)19-s + (−1.39e14 + 1.39e14i)21-s + 1.97e14i·23-s + 3.93e14i·25-s + (−1.49e15 + 1.49e15i)27-s + (−9.11e14 − 9.11e14i)29-s + 4.52e15·31-s + ⋯
L(s)  = 1  + (1.24 + 1.24i)3-s + (0.296 − 0.296i)5-s + 1.46i·7-s + 2.11i·9-s + (−1.05 + 1.05i)11-s + (0.133 + 0.133i)13-s + 0.739·15-s − 1.48·17-s + (1.11 + 1.11i)19-s + (−1.82 + 1.82i)21-s + 0.992i·23-s + 0.824i·25-s + (−1.39 + 1.39i)27-s + (−0.402 − 0.402i)29-s + 0.990·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.812 + 0.583i$
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: \((2,\ 64,\ (\ :21/2),\ -0.812 + 0.583i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.967708495\)
\(L(\frac12)\) \(\approx\) \(2.967708495\)
\(L(\frac{23}{2})\) 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 \( 1 + (-1.27e5 - 1.27e5i)T + 1.04e10iT^{2} \)
5 \( 1 + (-6.46e6 + 6.46e6i)T - 4.76e14iT^{2} \)
7 \( 1 - 1.09e9iT - 5.58e17T^{2} \)
11 \( 1 + (9.04e10 - 9.04e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-6.65e10 - 6.65e10i)T + 2.47e23iT^{2} \)
17 \( 1 + 1.23e13T + 6.90e25T^{2} \)
19 \( 1 + (-2.96e13 - 2.96e13i)T + 7.14e26iT^{2} \)
23 \( 1 - 1.97e14iT - 3.94e28T^{2} \)
29 \( 1 + (9.11e14 + 9.11e14i)T + 5.13e30iT^{2} \)
31 \( 1 - 4.52e15T + 2.08e31T^{2} \)
37 \( 1 + (-2.37e16 + 2.37e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 4.95e16iT - 7.38e33T^{2} \)
43 \( 1 + (-5.80e16 + 5.80e16i)T - 2.00e34iT^{2} \)
47 \( 1 + 1.35e17T + 1.30e35T^{2} \)
53 \( 1 + (-1.22e18 + 1.22e18i)T - 1.62e36iT^{2} \)
59 \( 1 + (-2.88e18 + 2.88e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (2.51e18 + 2.51e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (1.76e18 + 1.76e18i)T + 2.22e38iT^{2} \)
71 \( 1 - 6.89e18iT - 7.52e38T^{2} \)
73 \( 1 - 2.30e19iT - 1.34e39T^{2} \)
79 \( 1 + 5.60e18T + 7.08e39T^{2} \)
83 \( 1 + (-5.63e18 - 5.63e18i)T + 1.99e40iT^{2} \)
89 \( 1 + 1.52e20iT - 8.65e40T^{2} \)
97 \( 1 + 2.96e20T + 5.27e41T^{2} \)
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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

−11.42270328608093568893980834548, −10.01966994301105838195486201644, −9.403851305583979272763607950610, −8.618958294596826001889655734363, −7.59882937426962857221229646655, −5.61034107418726326614230872599, −4.85449828486076763104638380890, −3.66525043680380061280775830457, −2.48764198946091324479650419276, −1.96036339085962489376415217679, 0.45074275604197913529173657634, 1.06416449148259531621641743339, 2.50679922493288316323181331367, 3.02724205245266683347689476675, 4.43431071521994571205140179771, 6.31861733268667399474907999158, 7.12333271527656570111338493030, 7.983846339671738946756420802432, 8.863851255152462766691728062046, 10.25766739510303303012696962715

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