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Properties

Label 2-2e4-16.13-c27-0-12
Degree $2$
Conductor $16$
Sign $0.564 + 0.825i$
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  + (3.87e3 − 1.09e4i)2-s + (−2.14e6 − 2.14e6i)3-s + (−1.04e8 − 8.45e7i)4-s + (1.53e9 − 1.53e9i)5-s + (−3.16e10 + 1.50e10i)6-s + 9.29e10i·7-s + (−1.32e12 + 8.10e11i)8-s + 1.55e12i·9-s + (−1.08e13 − 2.27e13i)10-s + (−6.85e13 + 6.85e13i)11-s + (4.20e13 + 4.04e14i)12-s + (−1.09e15 − 1.09e15i)13-s + (1.01e15 + 3.60e14i)14-s − 6.57e15·15-s + (3.70e15 + 1.76e16i)16-s + 5.24e16·17-s + ⋯
L(s)  = 1  + (0.334 − 0.942i)2-s + (−0.775 − 0.775i)3-s + (−0.776 − 0.630i)4-s + (0.562 − 0.562i)5-s + (−0.990 + 0.471i)6-s + 0.362i·7-s + (−0.853 + 0.521i)8-s + 0.203i·9-s + (−0.342 − 0.718i)10-s + (−0.598 + 0.598i)11-s + (0.113 + 1.09i)12-s + (−0.999 − 0.999i)13-s + (0.341 + 0.121i)14-s − 0.872·15-s + (0.205 + 0.978i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.564 + 0.825i$
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: \((2,\ 16,\ (\ :27/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(14)\) \(\approx\) \(1.207615135\)
\(L(\frac12)\) \(\approx\) \(1.207615135\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.87e3 + 1.09e4i)T \)
good3 \( 1 + (2.14e6 + 2.14e6i)T + 7.62e12iT^{2} \)
5 \( 1 + (-1.53e9 + 1.53e9i)T - 7.45e18iT^{2} \)
7 \( 1 - 9.29e10iT - 6.57e22T^{2} \)
11 \( 1 + (6.85e13 - 6.85e13i)T - 1.31e28iT^{2} \)
13 \( 1 + (1.09e15 + 1.09e15i)T + 1.19e30iT^{2} \)
17 \( 1 - 5.24e16T + 1.66e33T^{2} \)
19 \( 1 + (-1.10e17 - 1.10e17i)T + 3.36e34iT^{2} \)
23 \( 1 - 2.41e18iT - 5.84e36T^{2} \)
29 \( 1 + (-3.79e19 - 3.79e19i)T + 3.05e39iT^{2} \)
31 \( 1 + 2.62e19T + 1.84e40T^{2} \)
37 \( 1 + (1.08e21 - 1.08e21i)T - 2.19e42iT^{2} \)
41 \( 1 + 4.76e21iT - 3.50e43T^{2} \)
43 \( 1 + (-9.21e21 + 9.21e21i)T - 1.26e44iT^{2} \)
47 \( 1 + 1.68e21T + 1.40e45T^{2} \)
53 \( 1 + (6.29e22 - 6.29e22i)T - 3.59e46iT^{2} \)
59 \( 1 + (7.91e23 - 7.91e23i)T - 6.50e47iT^{2} \)
61 \( 1 + (6.67e23 + 6.67e23i)T + 1.59e48iT^{2} \)
67 \( 1 + (3.91e24 + 3.91e24i)T + 2.01e49iT^{2} \)
71 \( 1 - 3.66e24iT - 9.63e49T^{2} \)
73 \( 1 + 2.35e24iT - 2.04e50T^{2} \)
79 \( 1 - 3.15e25T + 1.72e51T^{2} \)
83 \( 1 + (-9.31e25 - 9.31e25i)T + 6.53e51iT^{2} \)
89 \( 1 + 2.63e26iT - 4.30e52T^{2} \)
97 \( 1 + 1.60e25T + 4.39e53T^{2} \)
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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

−12.33269836989662281258504520160, −12.26540959432471022853037808192, −10.44308021476296877815869146133, −9.396073310656457900374302960697, −7.56914619556892848955171713876, −5.63835062414147678606913865351, −5.20896005015662794926260790204, −3.16350156881937103193129127171, −1.74532930600629333452624398630, −0.867411249936093623985480726274, 0.36984222389893286625630314997, 2.77336866124200143368850167166, 4.36444919591846181603385218167, 5.33266993203777442105300665117, 6.42094311887759022224465687105, 7.73499733802200691347321689624, 9.527753318223561471567586835045, 10.54683001958876509009885528770, 12.05717822230937319248976171182, 13.72411278437123224598515039144

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