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Properties

Label 2-2e4-16.3-c6-0-6
Degree $2$
Conductor $16$
Sign $0.985 + 0.168i$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.06 + 6.88i)2-s + (23.4 − 23.4i)3-s + (−30.9 − 56.0i)4-s + (55.3 − 55.3i)5-s + (66.2 + 257. i)6-s + 386.·7-s + (511. + 14.8i)8-s − 373. i·9-s + (156. + 605. i)10-s + (−208. − 208. i)11-s + (−2.04e3 − 589. i)12-s + (−1.97e3 − 1.97e3i)13-s + (−1.57e3 + 2.66e3i)14-s − 2.59e3i·15-s + (−2.18e3 + 3.46e3i)16-s + 8.60e3·17-s + ⋯
L(s)  = 1  + (−0.508 + 0.861i)2-s + (0.869 − 0.869i)3-s + (−0.483 − 0.875i)4-s + (0.442 − 0.442i)5-s + (0.306 + 1.19i)6-s + 1.12·7-s + (0.999 + 0.0289i)8-s − 0.512i·9-s + (0.156 + 0.605i)10-s + (−0.156 − 0.156i)11-s + (−1.18 − 0.341i)12-s + (−0.898 − 0.898i)13-s + (−0.572 + 0.969i)14-s − 0.769i·15-s + (−0.533 + 0.846i)16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ 0.985 + 0.168i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.54069 - 0.130919i\)
\(L(\frac12)\) \(\approx\) \(1.54069 - 0.130919i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.06 - 6.88i)T \)
good3 \( 1 + (-23.4 + 23.4i)T - 729iT^{2} \)
5 \( 1 + (-55.3 + 55.3i)T - 1.56e4iT^{2} \)
7 \( 1 - 386.T + 1.17e5T^{2} \)
11 \( 1 + (208. + 208. i)T + 1.77e6iT^{2} \)
13 \( 1 + (1.97e3 + 1.97e3i)T + 4.82e6iT^{2} \)
17 \( 1 - 8.60e3T + 2.41e7T^{2} \)
19 \( 1 + (7.11e3 - 7.11e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.45e4T + 1.48e8T^{2} \)
29 \( 1 + (4.50e3 + 4.50e3i)T + 5.94e8iT^{2} \)
31 \( 1 - 4.47e4iT - 8.87e8T^{2} \)
37 \( 1 + (-1.49e4 + 1.49e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 1.60e4iT - 4.75e9T^{2} \)
43 \( 1 + (-1.80e4 - 1.80e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 4.97e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.03e5 + 1.03e5i)T - 2.21e10iT^{2} \)
59 \( 1 + (-1.03e5 - 1.03e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (2.61e5 + 2.61e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (8.26e4 - 8.26e4i)T - 9.04e10iT^{2} \)
71 \( 1 - 2.15e4T + 1.28e11T^{2} \)
73 \( 1 - 4.31e5iT - 1.51e11T^{2} \)
79 \( 1 - 1.22e5iT - 2.43e11T^{2} \)
83 \( 1 + (8.04e4 - 8.04e4i)T - 3.26e11iT^{2} \)
89 \( 1 + 1.23e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.88e5T + 8.32e11T^{2} \)
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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

−17.78465734341468977975310831215, −16.66579996348299810869098328063, −14.75301300078723460808448791205, −14.10500967203076279663893299743, −12.61848651022666539234868754310, −10.15652672531022139430123011235, −8.359341320128431510400887735476, −7.65356107373848625345868534684, −5.42975687560348375978958748418, −1.57306759208670133936484453482, 2.35717262296360716883461784410, 4.34734351330968551524222713131, 7.919165746855298606703682047397, 9.381329675587900396180199857098, 10.41386232969749762277500202463, 11.95526035656651233460997289874, 13.98291578419681645579150108036, 14.84101090405571147290846504151, 16.75970146788844917359975628010, 17.99787658461983422415441992501

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