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Properties

Label 2-2e4-16.11-c6-0-1
Degree $2$
Conductor $16$
Sign $-0.611 - 0.790i$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.99 + 6.93i)2-s + (31.6 + 31.6i)3-s + (−32.1 − 55.3i)4-s + (69.9 + 69.9i)5-s + (−346. + 93.0i)6-s − 445.·7-s + (511. − 1.51i)8-s + 1.27e3i·9-s + (−764. + 205. i)10-s + (365. − 365. i)11-s + (736. − 2.77e3i)12-s + (1.78e3 − 1.78e3i)13-s + (1.77e3 − 3.08e3i)14-s + 4.43e3i·15-s + (−2.03e3 + 3.55e3i)16-s + 2.87e3·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (1.17 + 1.17i)3-s + (−0.501 − 0.865i)4-s + (0.559 + 0.559i)5-s + (−1.60 + 0.431i)6-s − 1.29·7-s + (0.999 − 0.00294i)8-s + 1.75i·9-s + (−0.764 + 0.205i)10-s + (0.274 − 0.274i)11-s + (0.426 − 1.60i)12-s + (0.813 − 0.813i)13-s + (0.647 − 1.12i)14-s + 1.31i·15-s + (−0.496 + 0.867i)16-s + 0.585·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.790i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.611 - 0.790i$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ -0.611 - 0.790i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.663431 + 1.35200i\)
\(L(\frac12)\) \(\approx\) \(0.663431 + 1.35200i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.99 - 6.93i)T \)
good3 \( 1 + (-31.6 - 31.6i)T + 729iT^{2} \)
5 \( 1 + (-69.9 - 69.9i)T + 1.56e4iT^{2} \)
7 \( 1 + 445.T + 1.17e5T^{2} \)
11 \( 1 + (-365. + 365. i)T - 1.77e6iT^{2} \)
13 \( 1 + (-1.78e3 + 1.78e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 2.87e3T + 2.41e7T^{2} \)
19 \( 1 + (-8.63e3 - 8.63e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 8.37e3T + 1.48e8T^{2} \)
29 \( 1 + (-1.38e4 + 1.38e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.06e4iT - 8.87e8T^{2} \)
37 \( 1 + (3.70e3 + 3.70e3i)T + 2.56e9iT^{2} \)
41 \( 1 + 4.58e4iT - 4.75e9T^{2} \)
43 \( 1 + (1.00e5 - 1.00e5i)T - 6.32e9iT^{2} \)
47 \( 1 + 3.67e4iT - 1.07e10T^{2} \)
53 \( 1 + (8.38e4 + 8.38e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (-4.31e4 + 4.31e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-7.90e4 + 7.90e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (1.51e5 + 1.51e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 3.83e5T + 1.28e11T^{2} \)
73 \( 1 + 1.01e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.80e5iT - 2.43e11T^{2} \)
83 \( 1 + (-4.39e4 - 4.39e4i)T + 3.26e11iT^{2} \)
89 \( 1 + 6.31e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.05e6T + 8.32e11T^{2} \)
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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

−18.28697487572840275447959538167, −16.42192460669580233264556404651, −15.73387815056333862267718059582, −14.46764357393952309431910695914, −13.57941403250593474246754547584, −10.17745011738480400585342741935, −9.688143527961812345048531110813, −8.148126435254602651269155218249, −6.01343910185102858702815457736, −3.43671713636622691714872819785, 1.35911019554165587882522998169, 3.15229677349882731285340120476, 7.00233420400668316031654740211, 8.766344861156987933666870948798, 9.622199371024602151864056583255, 12.08323177280129973040015689611, 13.20735053823140760336966844848, 13.83593784812234085920870899009, 16.26680253360406893694522834196, 17.85268703461581126287136214432

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