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Properties

Label 2-2e4-16.11-c10-0-5
Degree $2$
Conductor $16$
Sign $0.736 + 0.676i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.61 − 30.5i)2-s + (−107. − 107. i)3-s + (−839. + 586. i)4-s + (3.43e3 + 3.43e3i)5-s + (−2.24e3 + 4.30e3i)6-s + 8.77e3·7-s + (2.59e4 + 1.99e4i)8-s − 3.60e4i·9-s + (7.18e4 − 1.37e5i)10-s + (−1.70e5 + 1.70e5i)11-s + (1.52e5 + 2.70e4i)12-s + (3.35e5 − 3.35e5i)13-s + (−8.43e4 − 2.67e5i)14-s − 7.35e5i·15-s + (3.59e5 − 9.84e5i)16-s + 2.18e6·17-s + ⋯
L(s)  = 1  + (−0.300 − 0.953i)2-s + (−0.440 − 0.440i)3-s + (−0.819 + 0.573i)4-s + (1.09 + 1.09i)5-s + (−0.288 + 0.553i)6-s + 0.521·7-s + (0.792 + 0.609i)8-s − 0.611i·9-s + (0.718 − 1.37i)10-s + (−1.06 + 1.06i)11-s + (0.614 + 0.108i)12-s + (0.903 − 0.903i)13-s + (−0.156 − 0.497i)14-s − 0.969i·15-s + (0.343 − 0.939i)16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.736 + 0.676i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.41095 - 0.549748i\)
\(L(\frac12)\) \(\approx\) \(1.41095 - 0.549748i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.61 + 30.5i)T \)
good3 \( 1 + (107. + 107. i)T + 5.90e4iT^{2} \)
5 \( 1 + (-3.43e3 - 3.43e3i)T + 9.76e6iT^{2} \)
7 \( 1 - 8.77e3T + 2.82e8T^{2} \)
11 \( 1 + (1.70e5 - 1.70e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (-3.35e5 + 3.35e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 2.18e6T + 2.01e12T^{2} \)
19 \( 1 + (-1.50e6 - 1.50e6i)T + 6.13e12iT^{2} \)
23 \( 1 - 5.06e6T + 4.14e13T^{2} \)
29 \( 1 + (-7.02e6 + 7.02e6i)T - 4.20e14iT^{2} \)
31 \( 1 + 4.44e6iT - 8.19e14T^{2} \)
37 \( 1 + (-5.14e7 - 5.14e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 2.23e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.18e8 + 1.18e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 4.89e7iT - 5.25e16T^{2} \)
53 \( 1 + (-3.17e8 - 3.17e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (8.63e8 - 8.63e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-8.31e6 + 8.31e6i)T - 7.13e17iT^{2} \)
67 \( 1 + (9.40e8 + 9.40e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 2.57e8T + 3.25e18T^{2} \)
73 \( 1 - 3.20e8iT - 4.29e18T^{2} \)
79 \( 1 + 4.63e8iT - 9.46e18T^{2} \)
83 \( 1 + (-2.45e9 - 2.45e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 8.67e9iT - 3.11e19T^{2} \)
97 \( 1 + 5.79e9T + 7.37e19T^{2} \)
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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

−17.26307318149966100602309897138, −14.86492450836095425135954159587, −13.48851647523292574649708422410, −12.24847049432525189004186105944, −10.72271096054958016320826769221, −9.822772829799045275775692786352, −7.62642731775470447720183031556, −5.63434310312537183274674537084, −2.96346563971087716652354158717, −1.28990166782930377013407871725, 1.09572637128873237551593320341, 4.96913676555105459603981475443, 5.74721432627913875294389518999, 8.110908998363487995845359505473, 9.393830053484623403024726376902, 10.84722138410191847183795360195, 13.21914363880368700734324548960, 14.08389034767251768990708206483, 16.15796173078333582504367607458, 16.51924347214096874046412667381

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