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Properties

Label 2-2e4-16.3-c12-0-5
Degree $2$
Conductor $16$
Sign $-0.710 - 0.704i$
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  + (49.5 + 40.5i)2-s + (543. − 543. i)3-s + (807. + 4.01e3i)4-s + (−1.28e4 + 1.28e4i)5-s + (4.89e4 − 4.87e3i)6-s − 1.88e5·7-s + (−1.22e5 + 2.31e5i)8-s − 5.90e4i·9-s + (−1.15e6 + 1.14e5i)10-s + (1.91e6 + 1.91e6i)11-s + (2.62e6 + 1.74e6i)12-s + (2.40e6 + 2.40e6i)13-s + (−9.31e6 − 7.62e6i)14-s + 1.39e7i·15-s + (−1.54e7 + 6.48e6i)16-s − 2.94e7·17-s + ⋯
L(s)  = 1  + (0.773 + 0.633i)2-s + (0.745 − 0.745i)3-s + (0.197 + 0.980i)4-s + (−0.819 + 0.819i)5-s + (1.04 − 0.104i)6-s − 1.59·7-s + (−0.468 + 0.883i)8-s − 0.111i·9-s + (−1.15 + 0.114i)10-s + (1.08 + 1.08i)11-s + (0.877 + 0.583i)12-s + (0.498 + 0.498i)13-s + (−1.23 − 1.01i)14-s + 1.22i·15-s + (−0.922 + 0.386i)16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.710 - 0.704i$
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: \((2,\ 16,\ (\ :6),\ -0.710 - 0.704i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.826868 + 2.00839i\)
\(L(\frac12)\) \(\approx\) \(0.826868 + 2.00839i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-49.5 - 40.5i)T \)
good3 \( 1 + (-543. + 543. i)T - 5.31e5iT^{2} \)
5 \( 1 + (1.28e4 - 1.28e4i)T - 2.44e8iT^{2} \)
7 \( 1 + 1.88e5T + 1.38e10T^{2} \)
11 \( 1 + (-1.91e6 - 1.91e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (-2.40e6 - 2.40e6i)T + 2.32e13iT^{2} \)
17 \( 1 + 2.94e7T + 5.82e14T^{2} \)
19 \( 1 + (-6.15e7 + 6.15e7i)T - 2.21e15iT^{2} \)
23 \( 1 + 8.36e7T + 2.19e16T^{2} \)
29 \( 1 + (1.60e8 + 1.60e8i)T + 3.53e17iT^{2} \)
31 \( 1 - 4.34e8iT - 7.87e17T^{2} \)
37 \( 1 + (-8.05e8 + 8.05e8i)T - 6.58e18iT^{2} \)
41 \( 1 + 1.01e9iT - 2.25e19T^{2} \)
43 \( 1 + (-3.23e9 - 3.23e9i)T + 3.99e19iT^{2} \)
47 \( 1 + 3.68e9iT - 1.16e20T^{2} \)
53 \( 1 + (-6.63e8 + 6.63e8i)T - 4.91e20iT^{2} \)
59 \( 1 + (-3.79e10 - 3.79e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (-5.47e10 - 5.47e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-5.09e9 + 5.09e9i)T - 8.18e21iT^{2} \)
71 \( 1 + 4.29e10T + 1.64e22T^{2} \)
73 \( 1 + 1.35e11iT - 2.29e22T^{2} \)
79 \( 1 - 2.25e11iT - 5.90e22T^{2} \)
83 \( 1 + (3.93e11 - 3.93e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 5.31e11iT - 2.46e23T^{2} \)
97 \( 1 - 8.08e11T + 6.93e23T^{2} \)
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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

−16.17470297418861422297165742207, −15.22405743124664811757605161328, −13.91708515013847974875826464255, −12.94397171464447811132436863205, −11.59708303898571632084072545829, −9.102650458393333882275748976502, −7.21451745933254800221683290566, −6.71791134185030238936472803579, −3.91736916174999632371369749963, −2.64061472071433337104003429679, 0.62593323151311080197936679191, 3.33309557980683937724099226450, 3.95470443150736718946833678061, 6.13701580236524376077627294202, 8.810147253485960274107073577671, 9.862868812220084564942512320490, 11.67850748623252270494715389629, 12.91694107796287003350495027622, 14.17273357240850782491174617897, 15.74532052686175500152893328667

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