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Properties

Label 2-2e4-16.13-c21-0-28
Degree $2$
Conductor $16$
Sign $0.739 + 0.672i$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (697. + 1.26e3i)2-s + (−3.46e4 − 3.46e4i)3-s + (−1.12e6 + 1.77e6i)4-s + (2.92e7 − 2.92e7i)5-s + (1.97e7 − 6.81e7i)6-s + 8.76e8i·7-s + (−3.03e9 − 1.89e8i)8-s − 8.05e9i·9-s + (5.75e10 + 1.67e10i)10-s + (3.68e10 − 3.68e10i)11-s + (1.00e11 − 2.24e10i)12-s + (2.07e11 + 2.07e11i)13-s + (−1.11e12 + 6.11e11i)14-s − 2.02e12·15-s + (−1.87e12 − 3.97e12i)16-s − 1.20e13·17-s + ⋯
L(s)  = 1  + (0.481 + 0.876i)2-s + (−0.338 − 0.338i)3-s + (−0.535 + 0.844i)4-s + (1.33 − 1.33i)5-s + (0.133 − 0.460i)6-s + 1.17i·7-s + (−0.998 − 0.0624i)8-s − 0.770i·9-s + (1.81 + 0.528i)10-s + (0.428 − 0.428i)11-s + (0.467 − 0.104i)12-s + (0.417 + 0.417i)13-s + (−1.02 + 0.565i)14-s − 0.907·15-s + (−0.426 − 0.904i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.739 + 0.672i$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ 0.739 + 0.672i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.312855483\)
\(L(\frac12)\) \(\approx\) \(2.312855483\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-697. - 1.26e3i)T \)
good3 \( 1 + (3.46e4 + 3.46e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (-2.92e7 + 2.92e7i)T - 4.76e14iT^{2} \)
7 \( 1 - 8.76e8iT - 5.58e17T^{2} \)
11 \( 1 + (-3.68e10 + 3.68e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-2.07e11 - 2.07e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 1.20e13T + 6.90e25T^{2} \)
19 \( 1 + (-5.87e11 - 5.87e11i)T + 7.14e26iT^{2} \)
23 \( 1 + 2.29e14iT - 3.94e28T^{2} \)
29 \( 1 + (1.67e15 + 1.67e15i)T + 5.13e30iT^{2} \)
31 \( 1 - 4.09e15T + 2.08e31T^{2} \)
37 \( 1 + (-1.19e16 + 1.19e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 1.31e16iT - 7.38e33T^{2} \)
43 \( 1 + (-3.62e16 + 3.62e16i)T - 2.00e34iT^{2} \)
47 \( 1 - 5.93e17T + 1.30e35T^{2} \)
53 \( 1 + (-4.78e17 + 4.78e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (6.96e17 - 6.96e17i)T - 1.54e37iT^{2} \)
61 \( 1 + (4.32e17 + 4.32e17i)T + 3.10e37iT^{2} \)
67 \( 1 + (1.79e19 + 1.79e19i)T + 2.22e38iT^{2} \)
71 \( 1 + 2.07e19iT - 7.52e38T^{2} \)
73 \( 1 - 6.08e17iT - 1.34e39T^{2} \)
79 \( 1 - 1.30e20T + 7.08e39T^{2} \)
83 \( 1 + (1.51e20 + 1.51e20i)T + 1.99e40iT^{2} \)
89 \( 1 + 5.83e20iT - 8.65e40T^{2} \)
97 \( 1 + 1.33e21T + 5.27e41T^{2} \)
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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

−13.83660722732808000080847551414, −12.84369160253633638630388418464, −11.94706654493738384591141711170, −9.151151155409191269711571577315, −8.766711173662962203638456549468, −6.34248584214585216642636435731, −5.81504384231662534176823749748, −4.44748527038702909930209908904, −2.21005618168441301377284721761, −0.56087273256177563871740234298, 1.44346969449104333091188645293, 2.62789387059340280386206557394, 4.11782452433703677071635245660, 5.63854170183398534502397232632, 6.93014295704086929316098437392, 9.587989984301020169077045763610, 10.57538672527006642902289793800, 11.10575044262025960884733354174, 13.40094669231390377132365904872, 13.82402548193009327372085315207

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