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Properties

Label 2-2e4-16.11-c10-0-14
Degree $2$
Conductor $16$
Sign $-0.0847 + 0.996i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.46 + 31.9i)2-s + (−46.0 − 46.0i)3-s + (−1.01e3 + 156. i)4-s + (1.93e3 + 1.93e3i)5-s + (1.35e3 − 1.58e3i)6-s − 2.59e4·7-s + (−7.49e3 − 3.18e4i)8-s − 5.48e4i·9-s + (−5.69e4 + 6.64e4i)10-s + (−1.23e4 + 1.23e4i)11-s + (5.38e4 + 3.93e4i)12-s + (3.98e5 − 3.98e5i)13-s + (−6.37e4 − 8.27e5i)14-s − 1.78e5i·15-s + (9.99e5 − 3.17e5i)16-s − 2.30e6·17-s + ⋯
L(s)  = 1  + (0.0768 + 0.997i)2-s + (−0.189 − 0.189i)3-s + (−0.988 + 0.153i)4-s + (0.618 + 0.618i)5-s + (0.174 − 0.203i)6-s − 1.54·7-s + (−0.228 − 0.973i)8-s − 0.928i·9-s + (−0.569 + 0.664i)10-s + (−0.0766 + 0.0766i)11-s + (0.216 + 0.158i)12-s + (1.07 − 1.07i)13-s + (−0.118 − 1.53i)14-s − 0.234i·15-s + (0.953 − 0.302i)16-s − 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.0847 + 0.996i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.0847 + 0.996i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.178111 - 0.193911i\)
\(L(\frac12)\) \(\approx\) \(0.178111 - 0.193911i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.46 - 31.9i)T \)
good3 \( 1 + (46.0 + 46.0i)T + 5.90e4iT^{2} \)
5 \( 1 + (-1.93e3 - 1.93e3i)T + 9.76e6iT^{2} \)
7 \( 1 + 2.59e4T + 2.82e8T^{2} \)
11 \( 1 + (1.23e4 - 1.23e4i)T - 2.59e10iT^{2} \)
13 \( 1 + (-3.98e5 + 3.98e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 2.30e6T + 2.01e12T^{2} \)
19 \( 1 + (3.56e5 + 3.56e5i)T + 6.13e12iT^{2} \)
23 \( 1 + 7.31e6T + 4.14e13T^{2} \)
29 \( 1 + (1.42e7 - 1.42e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 3.35e7iT - 8.19e14T^{2} \)
37 \( 1 + (1.40e7 + 1.40e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.71e7iT - 1.34e16T^{2} \)
43 \( 1 + (1.08e8 - 1.08e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 2.53e8iT - 5.25e16T^{2} \)
53 \( 1 + (-1.86e8 - 1.86e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (4.10e8 - 4.10e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-8.65e8 + 8.65e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (1.18e9 + 1.18e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 8.95e8T + 3.25e18T^{2} \)
73 \( 1 - 1.21e8iT - 4.29e18T^{2} \)
79 \( 1 + 3.49e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.04e8 + 1.04e8i)T + 1.55e19iT^{2} \)
89 \( 1 - 8.97e9iT - 3.11e19T^{2} \)
97 \( 1 - 4.31e9T + 7.37e19T^{2} \)
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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

−16.13976408333563552492489721224, −15.21957264582008188549473801484, −13.60930112150454434414615890831, −12.70275534195356665786493157237, −10.26639214236486878653223421864, −8.929151621212483971908821820159, −6.73379292791318902354664997515, −6.04891545904354332558089175403, −3.44979287541940790367684248795, −0.11801957548725933109097721674, 2.05113469389871223595111778205, 4.12905266722040233725397981647, 5.99984530426124405783955787007, 8.876482603591588538336760615914, 9.950217865231319647795085640147, 11.37845352258176994682564605610, 13.14540316551942478079226789770, 13.54712699343017421824113234074, 15.95633721206735898295619704164, 17.03548537855030832470568145300

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