Properties

Label 2-2e4-16.5-c11-0-18
Degree $2$
Conductor $16$
Sign $-0.866 - 0.499i$
Analytic cond. $12.2934$
Root an. cond. $3.50620$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−30.9 − 33.0i)2-s + (585. − 585. i)3-s + (−133. + 2.04e3i)4-s + (−6.85e3 − 6.85e3i)5-s + (−3.74e4 − 1.22e3i)6-s + 1.99e4i·7-s + (7.16e4 − 5.88e4i)8-s − 5.08e5i·9-s + (−1.43e4 + 4.38e5i)10-s + (−9.64e4 − 9.64e4i)11-s + (1.11e6 + 1.27e6i)12-s + (−6.74e5 + 6.74e5i)13-s + (6.57e5 − 6.16e5i)14-s − 8.03e6·15-s + (−4.15e6 − 5.46e5i)16-s + 3.70e6·17-s + ⋯
L(s)  = 1  + (−0.683 − 0.729i)2-s + (1.39 − 1.39i)3-s + (−0.0652 + 0.997i)4-s + (−0.981 − 0.981i)5-s + (−1.96 − 0.0642i)6-s + 0.447i·7-s + (0.772 − 0.634i)8-s − 2.87i·9-s + (−0.0453 + 1.38i)10-s + (−0.180 − 0.180i)11-s + (1.29 + 1.47i)12-s + (−0.503 + 0.503i)13-s + (0.326 − 0.306i)14-s − 2.73·15-s + (−0.991 − 0.130i)16-s + 0.633·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(12.2934\)
Root analytic conductor: \(3.50620\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :11/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.323161 + 1.20673i\)
\(L(\frac12)\) \(\approx\) \(0.323161 + 1.20673i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (30.9 + 33.0i)T \)
good3 \( 1 + (-585. + 585. i)T - 1.77e5iT^{2} \)
5 \( 1 + (6.85e3 + 6.85e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 1.99e4iT - 1.97e9T^{2} \)
11 \( 1 + (9.64e4 + 9.64e4i)T + 2.85e11iT^{2} \)
13 \( 1 + (6.74e5 - 6.74e5i)T - 1.79e12iT^{2} \)
17 \( 1 - 3.70e6T + 3.42e13T^{2} \)
19 \( 1 + (2.61e6 - 2.61e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 2.97e7iT - 9.52e14T^{2} \)
29 \( 1 + (-4.14e7 + 4.14e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 6.07e7T + 2.54e16T^{2} \)
37 \( 1 + (4.37e8 + 4.37e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 1.20e9iT - 5.50e17T^{2} \)
43 \( 1 + (1.45e8 + 1.45e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.52e9T + 2.47e18T^{2} \)
53 \( 1 + (2.52e9 + 2.52e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (4.54e9 + 4.54e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-9.96e7 + 9.96e7i)T - 4.35e19iT^{2} \)
67 \( 1 + (-7.32e9 + 7.32e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.31e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.59e9iT - 3.13e20T^{2} \)
79 \( 1 + 4.08e9T + 7.47e20T^{2} \)
83 \( 1 + (-3.03e10 + 3.03e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 5.68e10iT - 2.77e21T^{2} \)
97 \( 1 + 5.73e10T + 7.15e21T^{2} \)
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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

−15.64199057645156149395812799438, −13.87348316300619107536043130641, −12.50063373807227503374166771646, −12.00941006338072447720172137799, −9.241823303791776572141905188733, −8.325278434336474833027695254909, −7.38932696691650370609757519566, −3.61665506502984511789911828194, −2.00081415741020161156402304292, −0.57503939181848003014329362622, 2.93811567187988018539479471950, 4.56142206897114557916900140381, 7.40308974551975897691961843387, 8.400972992053731004542324916463, 9.989480941281818736622296361774, 10.76654396898039693282678046114, 14.01507916185653770540552906523, 14.93723681412329129132574511194, 15.54622489915309019908774449605, 16.73717181730721466730860877238

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