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Properties

Label 2-2e4-16.11-c12-0-16
Degree $2$
Conductor $16$
Sign $0.694 - 0.719i$
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  + (63.7 + 5.98i)2-s + (652. + 652. i)3-s + (4.02e3 + 763. i)4-s + (−7.91e3 − 7.91e3i)5-s + (3.76e4 + 4.55e4i)6-s + 1.82e5·7-s + (2.51e5 + 7.27e4i)8-s + 3.20e5i·9-s + (−4.57e5 − 5.52e5i)10-s + (2.73e4 − 2.73e4i)11-s + (2.12e6 + 3.12e6i)12-s + (−5.51e6 + 5.51e6i)13-s + (1.16e7 + 1.09e6i)14-s − 1.03e7i·15-s + (1.56e7 + 6.14e6i)16-s − 9.08e6·17-s + ⋯
L(s)  = 1  + (0.995 + 0.0935i)2-s + (0.895 + 0.895i)3-s + (0.982 + 0.186i)4-s + (−0.506 − 0.506i)5-s + (0.807 + 0.975i)6-s + 1.55·7-s + (0.960 + 0.277i)8-s + 0.603i·9-s + (−0.457 − 0.552i)10-s + (0.0154 − 0.0154i)11-s + (0.712 + 1.04i)12-s + (−1.14 + 1.14i)13-s + (1.54 + 0.145i)14-s − 0.907i·15-s + (0.930 + 0.366i)16-s − 0.376·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.694 - 0.719i$
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: \((2,\ 16,\ (\ :6),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(4.23964 + 1.80078i\)
\(L(\frac12)\) \(\approx\) \(4.23964 + 1.80078i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-63.7 - 5.98i)T \)
good3 \( 1 + (-652. - 652. i)T + 5.31e5iT^{2} \)
5 \( 1 + (7.91e3 + 7.91e3i)T + 2.44e8iT^{2} \)
7 \( 1 - 1.82e5T + 1.38e10T^{2} \)
11 \( 1 + (-2.73e4 + 2.73e4i)T - 3.13e12iT^{2} \)
13 \( 1 + (5.51e6 - 5.51e6i)T - 2.32e13iT^{2} \)
17 \( 1 + 9.08e6T + 5.82e14T^{2} \)
19 \( 1 + (5.30e6 + 5.30e6i)T + 2.21e15iT^{2} \)
23 \( 1 - 8.24e7T + 2.19e16T^{2} \)
29 \( 1 + (5.25e8 - 5.25e8i)T - 3.53e17iT^{2} \)
31 \( 1 + 1.04e9iT - 7.87e17T^{2} \)
37 \( 1 + (3.53e9 + 3.53e9i)T + 6.58e18iT^{2} \)
41 \( 1 + 2.81e9iT - 2.25e19T^{2} \)
43 \( 1 + (-5.67e9 + 5.67e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 1.79e10iT - 1.16e20T^{2} \)
53 \( 1 + (1.91e10 + 1.91e10i)T + 4.91e20iT^{2} \)
59 \( 1 + (3.17e10 - 3.17e10i)T - 1.77e21iT^{2} \)
61 \( 1 + (-1.08e10 + 1.08e10i)T - 2.65e21iT^{2} \)
67 \( 1 + (5.50e10 + 5.50e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 5.61e10T + 1.64e22T^{2} \)
73 \( 1 - 9.77e10iT - 2.29e22T^{2} \)
79 \( 1 - 3.19e10iT - 5.90e22T^{2} \)
83 \( 1 + (-3.63e11 - 3.63e11i)T + 1.06e23iT^{2} \)
89 \( 1 + 6.91e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.28e11T + 6.93e23T^{2} \)
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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

−15.84499460719005120137434779296, −14.74157578027620566646363548196, −14.14896457273950221688798766799, −12.23492952278812048305530680464, −10.99056718350662808358591962745, −8.965337638678329899388941506373, −7.53315442861476885204686065391, −4.89666235785950243900573904333, −4.07433402216318529140384445441, −2.12331292762713194668371666897, 1.66130407142888369207326871183, 2.98683991734965526339651748878, 4.96674581421804452602013271810, 7.22558864503016042353207799206, 8.045566487620880248392494440394, 10.81104197518915509925551777029, 12.09612890241255782736251382080, 13.42030435135266383999390264131, 14.61093114029269593681845210613, 15.15906330033312244517122513942

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