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Properties

Label 2-2e4-16.13-c27-0-20
Degree $2$
Conductor $16$
Sign $0.853 + 0.521i$
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  + (−2.70e3 + 1.12e4i)2-s + (−3.16e6 − 3.16e6i)3-s + (−1.19e8 − 6.08e7i)4-s + (−9.16e8 + 9.16e8i)5-s + (4.42e10 − 2.71e10i)6-s − 2.63e11i·7-s + (1.00e12 − 1.18e12i)8-s + 1.24e13i·9-s + (−7.85e12 − 1.28e13i)10-s + (1.27e14 − 1.27e14i)11-s + (1.86e14 + 5.71e14i)12-s + (4.89e13 + 4.89e13i)13-s + (2.96e15 + 7.11e14i)14-s + 5.80e15·15-s + (1.06e16 + 1.45e16i)16-s − 1.67e16·17-s + ⋯
L(s)  = 1  + (−0.233 + 0.972i)2-s + (−1.14 − 1.14i)3-s + (−0.891 − 0.453i)4-s + (−0.335 + 0.335i)5-s + (1.38 − 0.847i)6-s − 1.02i·7-s + (0.648 − 0.761i)8-s + 1.62i·9-s + (−0.248 − 0.404i)10-s + (1.11 − 1.11i)11-s + (0.502 + 1.54i)12-s + (0.0448 + 0.0448i)13-s + (0.999 + 0.239i)14-s + 0.770·15-s + (0.588 + 0.808i)16-s − 0.411·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.853 + 0.521i$
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: \((2,\ 16,\ (\ :27/2),\ 0.853 + 0.521i)\)

Particular Values

\(L(14)\) \(\approx\) \(0.9261744993\)
\(L(\frac12)\) \(\approx\) \(0.9261744993\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.70e3 - 1.12e4i)T \)
good3 \( 1 + (3.16e6 + 3.16e6i)T + 7.62e12iT^{2} \)
5 \( 1 + (9.16e8 - 9.16e8i)T - 7.45e18iT^{2} \)
7 \( 1 + 2.63e11iT - 6.57e22T^{2} \)
11 \( 1 + (-1.27e14 + 1.27e14i)T - 1.31e28iT^{2} \)
13 \( 1 + (-4.89e13 - 4.89e13i)T + 1.19e30iT^{2} \)
17 \( 1 + 1.67e16T + 1.66e33T^{2} \)
19 \( 1 + (1.31e17 + 1.31e17i)T + 3.36e34iT^{2} \)
23 \( 1 - 3.03e18iT - 5.84e36T^{2} \)
29 \( 1 + (-5.07e19 - 5.07e19i)T + 3.05e39iT^{2} \)
31 \( 1 - 1.15e20T + 1.84e40T^{2} \)
37 \( 1 + (1.72e21 - 1.72e21i)T - 2.19e42iT^{2} \)
41 \( 1 + 4.98e21iT - 3.50e43T^{2} \)
43 \( 1 + (-1.24e22 + 1.24e22i)T - 1.26e44iT^{2} \)
47 \( 1 - 2.25e22T + 1.40e45T^{2} \)
53 \( 1 + (-1.09e23 + 1.09e23i)T - 3.59e46iT^{2} \)
59 \( 1 + (-4.56e23 + 4.56e23i)T - 6.50e47iT^{2} \)
61 \( 1 + (-1.24e24 - 1.24e24i)T + 1.59e48iT^{2} \)
67 \( 1 + (-1.56e24 - 1.56e24i)T + 2.01e49iT^{2} \)
71 \( 1 - 3.51e24iT - 9.63e49T^{2} \)
73 \( 1 - 2.24e25iT - 2.04e50T^{2} \)
79 \( 1 + 4.59e25T + 1.72e51T^{2} \)
83 \( 1 + (-4.56e25 - 4.56e25i)T + 6.53e51iT^{2} \)
89 \( 1 + 2.12e26iT - 4.30e52T^{2} \)
97 \( 1 - 6.87e26T + 4.39e53T^{2} \)
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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

−13.26148154185667407769155624682, −11.66150172616573217318358111371, −10.61234676340945623701353811133, −8.639841076205709328532987886368, −7.12191107527070477830316878000, −6.72016396187780881171999186085, −5.46675042132031623912795765353, −3.88715036398401306413961376906, −1.24476255522736685340585146496, −0.56308308266378782693149875208, 0.63675391437610185963414441004, 2.26701798500367960980060987519, 4.12468263878323037458248734119, 4.66164421518737569439066729525, 6.16903874714335691100423018299, 8.578873388861248349733317199197, 9.648104022119705013817284908266, 10.66067216875668274899673370655, 11.99414076220771820109827826391, 12.31189990327391707802023853376

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