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Properties

Label 2-2e4-16.5-c27-0-49
Degree $2$
Conductor $16$
Sign $-0.976 + 0.217i$
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  + (−6.84e3 + 9.34e3i)2-s + (2.95e6 − 2.95e6i)3-s + (−4.04e7 − 1.27e8i)4-s + (6.75e8 + 6.75e8i)5-s + (7.38e9 + 4.78e10i)6-s − 3.29e11i·7-s + (1.47e12 + 4.98e11i)8-s − 9.87e12i·9-s + (−1.09e13 + 1.68e12i)10-s + (−1.51e13 − 1.51e13i)11-s + (−4.98e14 − 2.58e14i)12-s + (−4.13e14 + 4.13e14i)13-s + (3.08e15 + 2.25e15i)14-s + 3.99e15·15-s + (−1.47e16 + 1.03e16i)16-s − 6.20e16·17-s + ⋯
L(s)  = 1  + (−0.591 + 0.806i)2-s + (1.07 − 1.07i)3-s + (−0.301 − 0.953i)4-s + (0.247 + 0.247i)5-s + (0.230 + 1.49i)6-s − 1.28i·7-s + (0.947 + 0.320i)8-s − 1.29i·9-s + (−0.346 + 0.0533i)10-s + (−0.132 − 0.132i)11-s + (−1.34 − 0.698i)12-s + (−0.378 + 0.378i)13-s + (1.03 + 0.760i)14-s + 0.530·15-s + (−0.818 + 0.574i)16-s − 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.976 + 0.217i$
Analytic conductor: \(73.8968\)
Root analytic conductor: \(8.59633\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :27/2),\ -0.976 + 0.217i)\)

Particular Values

\(L(14)\) \(\approx\) \(1.051214538\)
\(L(\frac12)\) \(\approx\) \(1.051214538\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.84e3 - 9.34e3i)T \)
good3 \( 1 + (-2.95e6 + 2.95e6i)T - 7.62e12iT^{2} \)
5 \( 1 + (-6.75e8 - 6.75e8i)T + 7.45e18iT^{2} \)
7 \( 1 + 3.29e11iT - 6.57e22T^{2} \)
11 \( 1 + (1.51e13 + 1.51e13i)T + 1.31e28iT^{2} \)
13 \( 1 + (4.13e14 - 4.13e14i)T - 1.19e30iT^{2} \)
17 \( 1 + 6.20e16T + 1.66e33T^{2} \)
19 \( 1 + (-1.52e17 + 1.52e17i)T - 3.36e34iT^{2} \)
23 \( 1 + 2.08e18iT - 5.84e36T^{2} \)
29 \( 1 + (1.58e19 - 1.58e19i)T - 3.05e39iT^{2} \)
31 \( 1 + 1.09e20T + 1.84e40T^{2} \)
37 \( 1 + (-7.19e20 - 7.19e20i)T + 2.19e42iT^{2} \)
41 \( 1 - 8.61e21iT - 3.50e43T^{2} \)
43 \( 1 + (-1.75e21 - 1.75e21i)T + 1.26e44iT^{2} \)
47 \( 1 - 1.47e22T + 1.40e45T^{2} \)
53 \( 1 + (2.01e23 + 2.01e23i)T + 3.59e46iT^{2} \)
59 \( 1 + (-9.87e22 - 9.87e22i)T + 6.50e47iT^{2} \)
61 \( 1 + (9.38e23 - 9.38e23i)T - 1.59e48iT^{2} \)
67 \( 1 + (2.27e24 - 2.27e24i)T - 2.01e49iT^{2} \)
71 \( 1 + 9.77e24iT - 9.63e49T^{2} \)
73 \( 1 - 1.86e25iT - 2.04e50T^{2} \)
79 \( 1 - 6.34e25T + 1.72e51T^{2} \)
83 \( 1 + (7.33e25 - 7.33e25i)T - 6.53e51iT^{2} \)
89 \( 1 + 3.71e26iT - 4.30e52T^{2} \)
97 \( 1 + 2.24e26T + 4.39e53T^{2} \)
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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

−13.19058048790582830387736299082, −10.91595797141228816666297575760, −9.481895978366492302383172897958, −8.293762817392725592149478336038, −7.20976543332528418700932656818, −6.59441407396138371219969352305, −4.48209474878273584713207372716, −2.58403065864523778335745081155, −1.37045908619897524653307093433, −0.24820173663156585387341877960, 1.85010374533680320129993083067, 2.74811792501876156401666587116, 3.86249718336451673249110985517, 5.27223966975023979359973382337, 7.80202490505116719541922113908, 9.116128401962582296322268033262, 9.384130606612039579463463760718, 10.83733565153397886868899581435, 12.28685940210181516974726991559, 13.61556909059887025212363532236

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