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Properties

Label 2-2e4-16.3-c12-0-0
Degree $2$
Conductor $16$
Sign $0.523 + 0.851i$
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  + (2.53 + 63.9i)2-s + (−322. + 322. i)3-s + (−4.08e3 + 324. i)4-s + (−1.64e4 + 1.64e4i)5-s + (−2.14e4 − 1.97e4i)6-s + 2.10e4·7-s + (−3.11e4 − 2.60e5i)8-s + 3.23e5i·9-s + (−1.09e6 − 1.01e6i)10-s + (−6.84e5 − 6.84e5i)11-s + (1.21e6 − 1.42e6i)12-s + (1.16e6 + 1.16e6i)13-s + (5.34e4 + 1.34e6i)14-s − 1.06e7i·15-s + (1.65e7 − 2.65e6i)16-s + 1.51e7·17-s + ⋯
L(s)  = 1  + (0.0396 + 0.999i)2-s + (−0.442 + 0.442i)3-s + (−0.996 + 0.0792i)4-s + (−1.05 + 1.05i)5-s + (−0.459 − 0.424i)6-s + 0.178·7-s + (−0.118 − 0.992i)8-s + 0.609i·9-s + (−1.09 − 1.01i)10-s + (−0.386 − 0.386i)11-s + (0.405 − 0.475i)12-s + (0.241 + 0.241i)13-s + (0.00709 + 0.178i)14-s − 0.931i·15-s + (0.987 − 0.158i)16-s + 0.626·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.523 + 0.851i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.523 + 0.851i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.139932 - 0.0782191i\)
\(L(\frac12)\) \(\approx\) \(0.139932 - 0.0782191i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.53 - 63.9i)T \)
good3 \( 1 + (322. - 322. i)T - 5.31e5iT^{2} \)
5 \( 1 + (1.64e4 - 1.64e4i)T - 2.44e8iT^{2} \)
7 \( 1 - 2.10e4T + 1.38e10T^{2} \)
11 \( 1 + (6.84e5 + 6.84e5i)T + 3.13e12iT^{2} \)
13 \( 1 + (-1.16e6 - 1.16e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 1.51e7T + 5.82e14T^{2} \)
19 \( 1 + (-1.09e7 + 1.09e7i)T - 2.21e15iT^{2} \)
23 \( 1 - 7.74e7T + 2.19e16T^{2} \)
29 \( 1 + (8.16e8 + 8.16e8i)T + 3.53e17iT^{2} \)
31 \( 1 - 1.34e9iT - 7.87e17T^{2} \)
37 \( 1 + (2.54e9 - 2.54e9i)T - 6.58e18iT^{2} \)
41 \( 1 - 5.12e8iT - 2.25e19T^{2} \)
43 \( 1 + (6.63e9 + 6.63e9i)T + 3.99e19iT^{2} \)
47 \( 1 + 1.21e10iT - 1.16e20T^{2} \)
53 \( 1 + (-1.64e10 + 1.64e10i)T - 4.91e20iT^{2} \)
59 \( 1 + (-4.80e10 - 4.80e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (3.19e10 + 3.19e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (2.65e10 - 2.65e10i)T - 8.18e21iT^{2} \)
71 \( 1 - 1.78e11T + 1.64e22T^{2} \)
73 \( 1 - 2.09e10iT - 2.29e22T^{2} \)
79 \( 1 - 3.44e11iT - 5.90e22T^{2} \)
83 \( 1 + (-2.33e11 + 2.33e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 8.93e11iT - 2.46e23T^{2} \)
97 \( 1 + 3.56e11T + 6.93e23T^{2} \)
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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

−16.91593611483642990233146210872, −15.84921656719059070735862542419, −14.96738391487647507065143707185, −13.61771319811136485119175263454, −11.58496880024975178785905824074, −10.30238032011644790988972792120, −8.211597468509231533939727911023, −7.00376817905787800457697207591, −5.23499103545068361661594065446, −3.59657281299215057935345337225, 0.07845922318698962088781138555, 1.26650905318697572717972738850, 3.69876137010580723858648982895, 5.24992450571154306768685160723, 7.82776715661873176153593349544, 9.281528256881872888665440294401, 11.18064600388064830381422217242, 12.25654757997444607864336832355, 12.96119214411735837108511616640, 14.93574262131787555714126991128

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