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Properties

Label 2-2e4-16.11-c12-0-19
Degree $2$
Conductor $16$
Sign $-0.909 - 0.416i$
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  + (−59.3 + 23.9i)2-s + (−234. − 234. i)3-s + (2.94e3 − 2.84e3i)4-s + (−1.81e4 − 1.81e4i)5-s + (1.95e4 + 8.31e3i)6-s + 1.12e5·7-s + (−1.06e5 + 2.39e5i)8-s − 4.21e5i·9-s + (1.51e6 + 6.43e5i)10-s + (1.85e6 − 1.85e6i)11-s + (−1.36e6 − 2.49e4i)12-s + (−4.76e6 + 4.76e6i)13-s + (−6.64e6 + 2.68e6i)14-s + 8.53e6i·15-s + (6.15e5 − 1.67e7i)16-s − 2.64e7·17-s + ⋯
L(s)  = 1  + (−0.927 + 0.374i)2-s + (−0.322 − 0.322i)3-s + (0.719 − 0.694i)4-s + (−1.16 − 1.16i)5-s + (0.419 + 0.178i)6-s + 0.952·7-s + (−0.407 + 0.912i)8-s − 0.792i·9-s + (1.51 + 0.643i)10-s + (1.04 − 1.04i)11-s + (−0.455 − 0.00836i)12-s + (−0.986 + 0.986i)13-s + (−0.883 + 0.356i)14-s + 0.749i·15-s + (0.0367 − 0.999i)16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.909 - 0.416i$
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: \((2,\ 16,\ (\ :6),\ -0.909 - 0.416i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0460214 + 0.211041i\)
\(L(\frac12)\) \(\approx\) \(0.0460214 + 0.211041i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (59.3 - 23.9i)T \)
good3 \( 1 + (234. + 234. i)T + 5.31e5iT^{2} \)
5 \( 1 + (1.81e4 + 1.81e4i)T + 2.44e8iT^{2} \)
7 \( 1 - 1.12e5T + 1.38e10T^{2} \)
11 \( 1 + (-1.85e6 + 1.85e6i)T - 3.13e12iT^{2} \)
13 \( 1 + (4.76e6 - 4.76e6i)T - 2.32e13iT^{2} \)
17 \( 1 + 2.64e7T + 5.82e14T^{2} \)
19 \( 1 + (1.10e7 + 1.10e7i)T + 2.21e15iT^{2} \)
23 \( 1 + 1.47e8T + 2.19e16T^{2} \)
29 \( 1 + (-6.85e8 + 6.85e8i)T - 3.53e17iT^{2} \)
31 \( 1 - 1.41e9iT - 7.87e17T^{2} \)
37 \( 1 + (-2.44e8 - 2.44e8i)T + 6.58e18iT^{2} \)
41 \( 1 - 4.41e9iT - 2.25e19T^{2} \)
43 \( 1 + (1.55e9 - 1.55e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 4.61e9iT - 1.16e20T^{2} \)
53 \( 1 + (4.04e8 + 4.04e8i)T + 4.91e20iT^{2} \)
59 \( 1 + (-2.96e9 + 2.96e9i)T - 1.77e21iT^{2} \)
61 \( 1 + (-9.42e9 + 9.42e9i)T - 2.65e21iT^{2} \)
67 \( 1 + (3.68e10 + 3.68e10i)T + 8.18e21iT^{2} \)
71 \( 1 - 5.60e10T + 1.64e22T^{2} \)
73 \( 1 + 1.89e10iT - 2.29e22T^{2} \)
79 \( 1 + 2.84e10iT - 5.90e22T^{2} \)
83 \( 1 + (3.85e11 + 3.85e11i)T + 1.06e23iT^{2} \)
89 \( 1 - 2.19e11iT - 2.46e23T^{2} \)
97 \( 1 - 8.48e10T + 6.93e23T^{2} \)
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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

−15.76593528581631683035543626983, −14.39090559897391657195840967708, −11.97255724248299854128633185356, −11.48574699242547729344207113229, −9.108315036639875492947033974879, −8.189803218038308304617558519347, −6.58224248861787232483545913155, −4.53003287990167360229832221836, −1.31962082161417588451489570853, −0.13320486199461078987439653295, 2.25459131835094605160349414022, 4.22931873749706665311172088361, 7.07570928167608120468375735499, 8.083645678719115580140825062803, 10.18130390322479283661944628857, 11.15843667911866278162148308701, 12.10538428861291307656095938095, 14.70216436115837970451681604632, 15.62433246877889079181608674819, 17.20885449865844438119178749093

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