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Properties

Label 2-2e4-16.13-c27-0-6
Degree $2$
Conductor $16$
Sign $0.959 - 0.283i$
Analytic cond. $73.8968$
Root an. cond. $8.59633$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15e4 − 305. i)2-s + (−2.33e6 − 2.33e6i)3-s + (1.34e8 + 7.08e6i)4-s + (−3.25e9 + 3.25e9i)5-s + (2.62e10 + 2.77e10i)6-s + 4.57e11i·7-s + (−1.55e12 − 1.23e11i)8-s + 3.25e12i·9-s + (3.86e13 − 3.67e13i)10-s + (7.31e13 − 7.31e13i)11-s + (−2.96e14 − 3.29e14i)12-s + (−1.05e15 − 1.05e15i)13-s + (1.40e14 − 5.30e15i)14-s + 1.51e16·15-s + (1.79e16 + 1.89e15i)16-s − 3.52e16·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0264i)2-s + (−0.844 − 0.844i)3-s + (0.998 + 0.0527i)4-s + (−1.19 + 1.19i)5-s + (0.822 + 0.866i)6-s + 1.78i·7-s + (−0.996 − 0.0791i)8-s + 0.426i·9-s + (1.22 − 1.16i)10-s + (0.638 − 0.638i)11-s + (−0.798 − 0.888i)12-s + (−0.962 − 0.962i)13-s + (0.0471 − 1.78i)14-s + 2.01·15-s + (0.994 + 0.105i)16-s − 0.863·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.959 - 0.283i$
Analytic conductor: \(73.8968\)
Root analytic conductor: \(8.59633\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :27/2),\ 0.959 - 0.283i)\)

Particular Values

\(L(14)\) \(\approx\) \(0.1882006014\)
\(L(\frac12)\) \(\approx\) \(0.1882006014\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15e4 + 305. i)T \)
good3 \( 1 + (2.33e6 + 2.33e6i)T + 7.62e12iT^{2} \)
5 \( 1 + (3.25e9 - 3.25e9i)T - 7.45e18iT^{2} \)
7 \( 1 - 4.57e11iT - 6.57e22T^{2} \)
11 \( 1 + (-7.31e13 + 7.31e13i)T - 1.31e28iT^{2} \)
13 \( 1 + (1.05e15 + 1.05e15i)T + 1.19e30iT^{2} \)
17 \( 1 + 3.52e16T + 1.66e33T^{2} \)
19 \( 1 + (1.46e17 + 1.46e17i)T + 3.36e34iT^{2} \)
23 \( 1 - 2.95e18iT - 5.84e36T^{2} \)
29 \( 1 + (3.27e19 + 3.27e19i)T + 3.05e39iT^{2} \)
31 \( 1 + 7.52e18T + 1.84e40T^{2} \)
37 \( 1 + (2.52e20 - 2.52e20i)T - 2.19e42iT^{2} \)
41 \( 1 + 3.98e21iT - 3.50e43T^{2} \)
43 \( 1 + (-1.03e22 + 1.03e22i)T - 1.26e44iT^{2} \)
47 \( 1 + 2.78e22T + 1.40e45T^{2} \)
53 \( 1 + (3.83e21 - 3.83e21i)T - 3.59e46iT^{2} \)
59 \( 1 + (5.02e23 - 5.02e23i)T - 6.50e47iT^{2} \)
61 \( 1 + (1.45e24 + 1.45e24i)T + 1.59e48iT^{2} \)
67 \( 1 + (3.20e24 + 3.20e24i)T + 2.01e49iT^{2} \)
71 \( 1 + 2.21e24iT - 9.63e49T^{2} \)
73 \( 1 - 8.58e24iT - 2.04e50T^{2} \)
79 \( 1 - 2.53e25T + 1.72e51T^{2} \)
83 \( 1 + (4.74e25 + 4.74e25i)T + 6.53e51iT^{2} \)
89 \( 1 - 7.83e25iT - 4.30e52T^{2} \)
97 \( 1 + 7.63e26T + 4.39e53T^{2} \)
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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

−12.42591968720588945767893030576, −11.65990684705579430198211076366, −10.99350180190961756847165892795, −9.082062564590294671818408537813, −7.76089135811001065754497909522, −6.72870912013469932812011328265, −5.78121145912201627209838685157, −3.14749591947077518739565059629, −2.10221468058390167469479308397, −0.28873841765962768660857300375, 0.25104264059367360516441378788, 1.48390514784318482086041793254, 4.14834797819416378377998876988, 4.54819389286975199719249383992, 6.75175119584775595745174149304, 7.84216501236718746561661029171, 9.281659758834378174727742466825, 10.46664465519401972903965033134, 11.40338182265039640239719724107, 12.50619780979329794711634531148

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