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Properties

Label 2-2e4-16.3-c8-0-8
Degree $2$
Conductor $16$
Sign $0.849 + 0.526i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.48 − 15.8i)2-s + (−100. + 100. i)3-s + (−243. + 78.4i)4-s + (623. − 623. i)5-s + (1.84e3 + 1.34e3i)6-s + 1.79e3·7-s + (1.84e3 + 3.65e3i)8-s − 1.37e4i·9-s + (−1.14e4 − 8.31e3i)10-s + (1.08e4 + 1.08e4i)11-s + (1.66e4 − 3.24e4i)12-s + (6.14e3 + 6.14e3i)13-s + (−4.46e3 − 2.84e4i)14-s + 1.25e5i·15-s + (5.32e4 − 3.82e4i)16-s + 6.09e4·17-s + ⋯
L(s)  = 1  + (−0.155 − 0.987i)2-s + (−1.24 + 1.24i)3-s + (−0.951 + 0.306i)4-s + (0.997 − 0.997i)5-s + (1.42 + 1.03i)6-s + 0.748·7-s + (0.450 + 0.892i)8-s − 2.10i·9-s + (−1.14 − 0.831i)10-s + (0.740 + 0.740i)11-s + (0.803 − 1.56i)12-s + (0.214 + 0.214i)13-s + (−0.116 − 0.739i)14-s + 2.48i·15-s + (0.812 − 0.583i)16-s + 0.729·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.526i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.849 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.849 + 0.526i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ 0.849 + 0.526i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.13789 - 0.324161i\)
\(L(\frac12)\) \(\approx\) \(1.13789 - 0.324161i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.48 + 15.8i)T \)
good3 \( 1 + (100. - 100. i)T - 6.56e3iT^{2} \)
5 \( 1 + (-623. + 623. i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.79e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.08e4 - 1.08e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-6.14e3 - 6.14e3i)T + 8.15e8iT^{2} \)
17 \( 1 - 6.09e4T + 6.97e9T^{2} \)
19 \( 1 + (-1.15e5 + 1.15e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 1.91e5T + 7.83e10T^{2} \)
29 \( 1 + (-6.99e5 - 6.99e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 2.98e5iT - 8.52e11T^{2} \)
37 \( 1 + (-7.22e5 + 7.22e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.33e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.62e6 - 2.62e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 4.39e6iT - 2.38e13T^{2} \)
53 \( 1 + (-4.09e6 + 4.09e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (8.81e5 + 8.81e5i)T + 1.46e14iT^{2} \)
61 \( 1 + (-2.76e6 - 2.76e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.48e7 + 1.48e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 4.27e7T + 6.45e14T^{2} \)
73 \( 1 - 4.22e7iT - 8.06e14T^{2} \)
79 \( 1 - 8.66e6iT - 1.51e15T^{2} \)
83 \( 1 + (2.44e7 - 2.44e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 8.38e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.23e7T + 7.83e15T^{2} \)
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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

−17.38394721758898584122232767098, −16.24905328743430770633452325039, −14.25628595578561024647599985956, −12.43418992278537434751621186800, −11.38029375768181020876199830711, −10.00423908598270830720308468763, −9.132993507433825102833092624523, −5.42414677085614038596087785532, −4.42386861033724575973856203704, −1.16016375852919684703625048604, 1.21818105017151394745080537305, 5.65048524159780183862300482199, 6.43936840223078489815864688434, 7.85159801219942583362366372760, 10.27465912665969098216585730087, 11.81349513030840739397238774568, 13.60115990087746106273888049806, 14.35760017965754257921194263340, 16.49190215352049993017425701665, 17.54957531026592182727415915038

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