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Properties

Label 2-2e4-16.11-c12-0-18
Degree $2$
Conductor $16$
Sign $0.0254 + 0.999i$
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  + (40.2 − 49.7i)2-s + (87.0 + 87.0i)3-s + (−850. − 4.00e3i)4-s + (8.11e3 + 8.11e3i)5-s + (7.83e3 − 822. i)6-s + 1.49e5·7-s + (−2.33e5 − 1.19e5i)8-s − 5.16e5i·9-s + (7.30e5 − 7.66e4i)10-s + (1.93e5 − 1.93e5i)11-s + (2.74e5 − 4.22e5i)12-s + (2.19e6 − 2.19e6i)13-s + (6.03e6 − 7.44e6i)14-s + 1.41e6i·15-s + (−1.53e7 + 6.81e6i)16-s + 3.62e6·17-s + ⋯
L(s)  = 1  + (0.629 − 0.777i)2-s + (0.119 + 0.119i)3-s + (−0.207 − 0.978i)4-s + (0.519 + 0.519i)5-s + (0.167 − 0.0176i)6-s + 1.27·7-s + (−0.890 − 0.454i)8-s − 0.971i·9-s + (0.730 − 0.0766i)10-s + (0.109 − 0.109i)11-s + (0.0920 − 0.141i)12-s + (0.454 − 0.454i)13-s + (0.801 − 0.989i)14-s + 0.124i·15-s + (−0.913 + 0.406i)16-s + 0.150·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.0254 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.0254 + 0.999i$
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: \((2,\ 16,\ (\ :6),\ 0.0254 + 0.999i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.18780 - 2.13281i\)
\(L(\frac12)\) \(\approx\) \(2.18780 - 2.13281i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-40.2 + 49.7i)T \)
good3 \( 1 + (-87.0 - 87.0i)T + 5.31e5iT^{2} \)
5 \( 1 + (-8.11e3 - 8.11e3i)T + 2.44e8iT^{2} \)
7 \( 1 - 1.49e5T + 1.38e10T^{2} \)
11 \( 1 + (-1.93e5 + 1.93e5i)T - 3.13e12iT^{2} \)
13 \( 1 + (-2.19e6 + 2.19e6i)T - 2.32e13iT^{2} \)
17 \( 1 - 3.62e6T + 5.82e14T^{2} \)
19 \( 1 + (5.40e7 + 5.40e7i)T + 2.21e15iT^{2} \)
23 \( 1 - 1.88e8T + 2.19e16T^{2} \)
29 \( 1 + (2.28e7 - 2.28e7i)T - 3.53e17iT^{2} \)
31 \( 1 - 1.07e9iT - 7.87e17T^{2} \)
37 \( 1 + (-2.33e9 - 2.33e9i)T + 6.58e18iT^{2} \)
41 \( 1 - 8.66e9iT - 2.25e19T^{2} \)
43 \( 1 + (-1.65e9 + 1.65e9i)T - 3.99e19iT^{2} \)
47 \( 1 + 1.96e9iT - 1.16e20T^{2} \)
53 \( 1 + (-1.44e9 - 1.44e9i)T + 4.91e20iT^{2} \)
59 \( 1 + (-9.36e9 + 9.36e9i)T - 1.77e21iT^{2} \)
61 \( 1 + (5.53e10 - 5.53e10i)T - 2.65e21iT^{2} \)
67 \( 1 + (5.91e10 + 5.91e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 9.64e10T + 1.64e22T^{2} \)
73 \( 1 - 1.82e11iT - 2.29e22T^{2} \)
79 \( 1 + 2.07e11iT - 5.90e22T^{2} \)
83 \( 1 + (2.54e11 + 2.54e11i)T + 1.06e23iT^{2} \)
89 \( 1 - 4.21e11iT - 2.46e23T^{2} \)
97 \( 1 - 1.03e12T + 6.93e23T^{2} \)
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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

−15.15792000461015793259631647255, −14.48442180832467691687604893405, −13.11261566423921649737723808990, −11.54350279314197719367877958775, −10.49565327693373489384834680111, −8.862784331320720524580188454158, −6.38829874762760560561063224666, −4.67799054174657549067556412961, −2.87349804745734631763866181756, −1.15801776080267477752067509263, 1.89851996162477337795217625308, 4.41169291033586680194064294532, 5.66836639847510305078508102781, 7.62562966728926052407479581414, 8.795874732758072645079322553953, 11.13034444902025051542467415874, 12.78510947377394576213930128805, 13.91119654743733586110279665667, 14.92589892433910656249737073318, 16.58540832166308809625522946175

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