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Properties

Label 2-2e6-16.13-c21-0-1
Degree $2$
Conductor $64$
Sign $-0.761 - 0.648i$
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  + (−1.25e5 − 1.25e5i)3-s + (−7.58e5 + 7.58e5i)5-s − 6.32e7i·7-s + 2.09e10i·9-s + (3.72e10 − 3.72e10i)11-s + (3.30e11 + 3.30e11i)13-s + 1.90e11·15-s − 1.09e13·17-s + (3.19e13 + 3.19e13i)19-s + (−7.92e12 + 7.92e12i)21-s + 1.76e14i·23-s + 4.75e14i·25-s + (1.31e15 − 1.31e15i)27-s + (1.65e15 + 1.65e15i)29-s + 1.21e15·31-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)3-s + (−0.0347 + 0.0347i)5-s − 0.0846i·7-s + 2.00i·9-s + (0.432 − 0.432i)11-s + (0.664 + 0.664i)13-s + 0.0851·15-s − 1.31·17-s + (1.19 + 1.19i)19-s + (−0.103 + 0.103i)21-s + 0.887i·23-s + 0.997i·25-s + (1.22 − 1.22i)27-s + (0.729 + 0.729i)29-s + 0.266·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.761 - 0.648i$
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: \((2,\ 64,\ (\ :21/2),\ -0.761 - 0.648i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.1635947072\)
\(L(\frac12)\) \(\approx\) \(0.1635947072\)
\(L(\frac{23}{2})\) 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 \( 1 + (1.25e5 + 1.25e5i)T + 1.04e10iT^{2} \)
5 \( 1 + (7.58e5 - 7.58e5i)T - 4.76e14iT^{2} \)
7 \( 1 + 6.32e7iT - 5.58e17T^{2} \)
11 \( 1 + (-3.72e10 + 3.72e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-3.30e11 - 3.30e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 1.09e13T + 6.90e25T^{2} \)
19 \( 1 + (-3.19e13 - 3.19e13i)T + 7.14e26iT^{2} \)
23 \( 1 - 1.76e14iT - 3.94e28T^{2} \)
29 \( 1 + (-1.65e15 - 1.65e15i)T + 5.13e30iT^{2} \)
31 \( 1 - 1.21e15T + 2.08e31T^{2} \)
37 \( 1 + (1.90e16 - 1.90e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 5.44e16iT - 7.38e33T^{2} \)
43 \( 1 + (-2.40e16 + 2.40e16i)T - 2.00e34iT^{2} \)
47 \( 1 + 3.90e17T + 1.30e35T^{2} \)
53 \( 1 + (2.83e17 - 2.83e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (-1.76e17 + 1.76e17i)T - 1.54e37iT^{2} \)
61 \( 1 + (6.75e18 + 6.75e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (1.61e19 + 1.61e19i)T + 2.22e38iT^{2} \)
71 \( 1 + 2.50e19iT - 7.52e38T^{2} \)
73 \( 1 - 1.80e19iT - 1.34e39T^{2} \)
79 \( 1 - 1.13e20T + 7.08e39T^{2} \)
83 \( 1 + (1.34e20 + 1.34e20i)T + 1.99e40iT^{2} \)
89 \( 1 + 2.40e20iT - 8.65e40T^{2} \)
97 \( 1 - 3.70e20T + 5.27e41T^{2} \)
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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

−11.52655491225972251961537975522, −10.68259572845960844570535106978, −9.079788288447192285274046159692, −7.74756535591258680541007789235, −6.77484619578334033719202931577, −6.03670893607722803369799790220, −4.99077469344966954569237513180, −3.47335981070068672353033837275, −1.72968085927467449827754890454, −1.19352936438193894340738915000, 0.04668339547877549722826579658, 0.920870864081468131029590438548, 2.75865692619535903943404874673, 4.16393916525018107416719867130, 4.79342994120813767631690501386, 5.92232450631661098680064920954, 6.82933356757215762205300546306, 8.623613464152493321380305957213, 9.656909296276981824992395898569, 10.59551394697408018547344380449

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