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Properties

Label 2-2e4-16.11-c6-0-7
Degree $2$
Conductor $16$
Sign $-0.815 + 0.579i$
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  + (−7.92 − 1.11i)2-s + (8.95 + 8.95i)3-s + (61.5 + 17.6i)4-s + (−127. − 127. i)5-s + (−60.9 − 80.8i)6-s − 458.·7-s + (−467. − 208. i)8-s − 568. i·9-s + (865. + 1.14e3i)10-s + (912. − 912. i)11-s + (392. + 708. i)12-s + (−1.92e3 + 1.92e3i)13-s + (3.63e3 + 512. i)14-s − 2.27e3i·15-s + (3.47e3 + 2.17e3i)16-s − 1.61e3·17-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.331 + 0.331i)3-s + (0.961 + 0.276i)4-s + (−1.01 − 1.01i)5-s + (−0.281 − 0.374i)6-s − 1.33·7-s + (−0.913 − 0.407i)8-s − 0.780i·9-s + (0.865 + 1.14i)10-s + (0.685 − 0.685i)11-s + (0.226 + 0.410i)12-s + (−0.876 + 0.876i)13-s + (1.32 + 0.186i)14-s − 0.674i·15-s + (0.847 + 0.531i)16-s − 0.329·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.815 + 0.579i$
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: \((2,\ 16,\ (\ :3),\ -0.815 + 0.579i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.113770 - 0.356433i\)
\(L(\frac12)\) \(\approx\) \(0.113770 - 0.356433i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.92 + 1.11i)T \)
good3 \( 1 + (-8.95 - 8.95i)T + 729iT^{2} \)
5 \( 1 + (127. + 127. i)T + 1.56e4iT^{2} \)
7 \( 1 + 458.T + 1.17e5T^{2} \)
11 \( 1 + (-912. + 912. i)T - 1.77e6iT^{2} \)
13 \( 1 + (1.92e3 - 1.92e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 1.61e3T + 2.41e7T^{2} \)
19 \( 1 + (784. + 784. i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.12e4T + 1.48e8T^{2} \)
29 \( 1 + (2.32e4 - 2.32e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.45e4iT - 8.87e8T^{2} \)
37 \( 1 + (-1.94e4 - 1.94e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 1.03e5iT - 4.75e9T^{2} \)
43 \( 1 + (4.42e4 - 4.42e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.68e5iT - 1.07e10T^{2} \)
53 \( 1 + (1.24e5 + 1.24e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (8.48e4 - 8.48e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.10e5 + 1.10e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-2.68e5 - 2.68e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 1.62e5T + 1.28e11T^{2} \)
73 \( 1 + 2.85e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.82e5iT - 2.43e11T^{2} \)
83 \( 1 + (3.04e4 + 3.04e4i)T + 3.26e11iT^{2} \)
89 \( 1 - 4.09e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.45e6T + 8.32e11T^{2} \)
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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

−16.92397488514224451686584844551, −16.24468232528715195338325696947, −15.09334121296663330968599975250, −12.67654765822137877728981508363, −11.60440409120746605881860726768, −9.515909136515368872774792636834, −8.777606515469921628887312880751, −6.83152516540770944025868210848, −3.61736216764498603811428040112, −0.33244283634086904706063069171, 2.88708472455068552461074071590, 6.78270163813573482122524860827, 7.75327480009997581799611019974, 9.676216411850673956951962348861, 11.02443822824402014758864955721, 12.61955780768232644345763021282, 14.74519187835831785549571684418, 15.70665046771436108205396264296, 17.06112432062277606480093194592, 18.64367396165595003389617473157

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