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Properties

Label 2-2e4-16.11-c12-0-8
Degree $2$
Conductor $16$
Sign $0.536 - 0.843i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−63.9 − 2.78i)2-s + (398. + 398. i)3-s + (4.08e3 + 355. i)4-s + (1.48e4 + 1.48e4i)5-s + (−2.43e4 − 2.65e4i)6-s + 1.13e5·7-s + (−2.59e5 − 3.40e4i)8-s − 2.14e5i·9-s + (−9.08e5 − 9.91e5i)10-s + (5.03e5 − 5.03e5i)11-s + (1.48e6 + 1.76e6i)12-s + (−3.27e5 + 3.27e5i)13-s + (−7.25e6 − 3.15e5i)14-s + 1.18e7i·15-s + (1.65e7 + 2.90e6i)16-s + 1.83e7·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0434i)2-s + (0.546 + 0.546i)3-s + (0.996 + 0.0868i)4-s + (0.950 + 0.950i)5-s + (−0.521 − 0.569i)6-s + 0.964·7-s + (−0.991 − 0.130i)8-s − 0.403i·9-s + (−0.908 − 0.991i)10-s + (0.284 − 0.284i)11-s + (0.496 + 0.591i)12-s + (−0.0679 + 0.0679i)13-s + (−0.963 − 0.0419i)14-s + 1.03i·15-s + (0.984 + 0.173i)16-s + 0.758·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.536 - 0.843i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.536 - 0.843i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.65219 + 0.907086i\)
\(L(\frac12)\) \(\approx\) \(1.65219 + 0.907086i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (63.9 + 2.78i)T \)
good3 \( 1 + (-398. - 398. i)T + 5.31e5iT^{2} \)
5 \( 1 + (-1.48e4 - 1.48e4i)T + 2.44e8iT^{2} \)
7 \( 1 - 1.13e5T + 1.38e10T^{2} \)
11 \( 1 + (-5.03e5 + 5.03e5i)T - 3.13e12iT^{2} \)
13 \( 1 + (3.27e5 - 3.27e5i)T - 2.32e13iT^{2} \)
17 \( 1 - 1.83e7T + 5.82e14T^{2} \)
19 \( 1 + (-4.69e7 - 4.69e7i)T + 2.21e15iT^{2} \)
23 \( 1 + 1.25e8T + 2.19e16T^{2} \)
29 \( 1 + (6.07e8 - 6.07e8i)T - 3.53e17iT^{2} \)
31 \( 1 + 5.99e8iT - 7.87e17T^{2} \)
37 \( 1 + (-2.83e9 - 2.83e9i)T + 6.58e18iT^{2} \)
41 \( 1 + 4.78e9iT - 2.25e19T^{2} \)
43 \( 1 + (7.09e9 - 7.09e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 7.56e9iT - 1.16e20T^{2} \)
53 \( 1 + (-1.49e10 - 1.49e10i)T + 4.91e20iT^{2} \)
59 \( 1 + (2.85e10 - 2.85e10i)T - 1.77e21iT^{2} \)
61 \( 1 + (-6.31e10 + 6.31e10i)T - 2.65e21iT^{2} \)
67 \( 1 + (9.82e10 + 9.82e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 1.46e11T + 1.64e22T^{2} \)
73 \( 1 + 2.56e11iT - 2.29e22T^{2} \)
79 \( 1 + 5.61e10iT - 5.90e22T^{2} \)
83 \( 1 + (-6.41e9 - 6.41e9i)T + 1.06e23iT^{2} \)
89 \( 1 + 1.56e11iT - 2.46e23T^{2} \)
97 \( 1 - 1.35e12T + 6.93e23T^{2} \)
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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

−16.56684021973995328331552714914, −14.91337471974906065339149865791, −14.23366291972699754208590427163, −11.73392685940203397488954902560, −10.32474278604151525097057028483, −9.393722155456276419130864495185, −7.81888875082696429954159027112, −6.06110723620106118420851020860, −3.20419879555169841624558838028, −1.58174640302838166459105177847, 1.16197378944557045799616682894, 2.16228829897164548875407493119, 5.39685943278294582101168265207, 7.47976831302473499269111914224, 8.623029974330167956123854317861, 9.860304572706647871202587850065, 11.64856371502030607931089934396, 13.24779758142803084641637537537, 14.56777216301868178736076857887, 16.35423159762621966456175309330

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