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Properties

Label 2-2e4-16.11-c12-0-4
Degree $2$
Conductor $16$
Sign $0.894 - 0.446i$
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  + (−62.9 − 11.3i)2-s + (−429. − 429. i)3-s + (3.83e3 + 1.43e3i)4-s + (−2.47e3 − 2.47e3i)5-s + (2.21e4 + 3.19e4i)6-s − 1.98e5·7-s + (−2.25e5 − 1.33e5i)8-s − 1.62e5i·9-s + (1.27e5 + 1.83e5i)10-s + (−1.13e6 + 1.13e6i)11-s + (−1.03e6 − 2.26e6i)12-s + (1.22e6 − 1.22e6i)13-s + (1.25e7 + 2.26e6i)14-s + 2.12e6i·15-s + (1.26e7 + 1.10e7i)16-s − 8.93e5·17-s + ⋯
L(s)  = 1  + (−0.984 − 0.177i)2-s + (−0.589 − 0.589i)3-s + (0.936 + 0.349i)4-s + (−0.158 − 0.158i)5-s + (0.474 + 0.684i)6-s − 1.68·7-s + (−0.859 − 0.510i)8-s − 0.305i·9-s + (0.127 + 0.183i)10-s + (−0.638 + 0.638i)11-s + (−0.345 − 0.758i)12-s + (0.253 − 0.253i)13-s + (1.66 + 0.300i)14-s + 0.186i·15-s + (0.755 + 0.655i)16-s − 0.0370·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.894 - 0.446i$
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: \((2,\ 16,\ (\ :6),\ 0.894 - 0.446i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.416894 + 0.0982667i\)
\(L(\frac12)\) \(\approx\) \(0.416894 + 0.0982667i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (62.9 + 11.3i)T \)
good3 \( 1 + (429. + 429. i)T + 5.31e5iT^{2} \)
5 \( 1 + (2.47e3 + 2.47e3i)T + 2.44e8iT^{2} \)
7 \( 1 + 1.98e5T + 1.38e10T^{2} \)
11 \( 1 + (1.13e6 - 1.13e6i)T - 3.13e12iT^{2} \)
13 \( 1 + (-1.22e6 + 1.22e6i)T - 2.32e13iT^{2} \)
17 \( 1 + 8.93e5T + 5.82e14T^{2} \)
19 \( 1 + (-2.55e7 - 2.55e7i)T + 2.21e15iT^{2} \)
23 \( 1 - 2.41e8T + 2.19e16T^{2} \)
29 \( 1 + (-1.87e8 + 1.87e8i)T - 3.53e17iT^{2} \)
31 \( 1 - 9.02e8iT - 7.87e17T^{2} \)
37 \( 1 + (1.02e9 + 1.02e9i)T + 6.58e18iT^{2} \)
41 \( 1 - 6.51e9iT - 2.25e19T^{2} \)
43 \( 1 + (-6.20e9 + 6.20e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 1.35e10iT - 1.16e20T^{2} \)
53 \( 1 + (-9.75e9 - 9.75e9i)T + 4.91e20iT^{2} \)
59 \( 1 + (2.24e10 - 2.24e10i)T - 1.77e21iT^{2} \)
61 \( 1 + (-4.44e9 + 4.44e9i)T - 2.65e21iT^{2} \)
67 \( 1 + (5.79e10 + 5.79e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 2.01e11T + 1.64e22T^{2} \)
73 \( 1 + 1.84e11iT - 2.29e22T^{2} \)
79 \( 1 - 2.43e11iT - 5.90e22T^{2} \)
83 \( 1 + (6.52e10 + 6.52e10i)T + 1.06e23iT^{2} \)
89 \( 1 - 2.19e11iT - 2.46e23T^{2} \)
97 \( 1 + 4.30e11T + 6.93e23T^{2} \)
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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

−16.46112284014806186779686186116, −15.49116809124672683799261171775, −12.89886660058528814360643408244, −12.16899075404600404120534024247, −10.44748692616591929418308446098, −9.196048043845393127505537058536, −7.28938385685353887312441615800, −6.17420673369026238624902876395, −3.04862623794833313947251118238, −0.864110767425548160888780558189, 0.36206082136846197622235073887, 3.02485845847540239702791494278, 5.65185009498984018831349520960, 7.11381581450246472906002721172, 9.079956069226377874283502265599, 10.29177682812322363061021734094, 11.33911054057968350239414790666, 13.21936730939385253126847345497, 15.47633318988635871918296617622, 16.19667649514913456995797493601

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