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Properties

Label 2-2e4-16.11-c10-0-1
Degree $2$
Conductor $16$
Sign $-0.518 + 0.855i$
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  + (−1.21 + 31.9i)2-s + (287. + 287. i)3-s + (−1.02e3 − 77.6i)4-s + (−3.97e3 − 3.97e3i)5-s + (−9.53e3 + 8.83e3i)6-s − 1.17e4·7-s + (3.72e3 − 3.25e4i)8-s + 1.05e5i·9-s + (1.31e5 − 1.22e5i)10-s + (−2.67e3 + 2.67e3i)11-s + (−2.70e5 − 3.15e5i)12-s + (−4.43e5 + 4.43e5i)13-s + (1.42e4 − 3.75e5i)14-s − 2.28e6i·15-s + (1.03e6 + 1.58e5i)16-s − 1.39e4·17-s + ⋯
L(s)  = 1  + (−0.0379 + 0.999i)2-s + (1.18 + 1.18i)3-s + (−0.997 − 0.0758i)4-s + (−1.27 − 1.27i)5-s + (−1.22 + 1.13i)6-s − 0.699·7-s + (0.113 − 0.993i)8-s + 1.79i·9-s + (1.31 − 1.22i)10-s + (−0.0165 + 0.0165i)11-s + (−1.08 − 1.26i)12-s + (−1.19 + 1.19i)13-s + (0.0265 − 0.698i)14-s − 3.00i·15-s + (0.988 + 0.151i)16-s − 0.00980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.518 + 0.855i$
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: \((2,\ 16,\ (\ :5),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.275753 - 0.489382i\)
\(L(\frac12)\) \(\approx\) \(0.275753 - 0.489382i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 - 31.9i)T \)
good3 \( 1 + (-287. - 287. i)T + 5.90e4iT^{2} \)
5 \( 1 + (3.97e3 + 3.97e3i)T + 9.76e6iT^{2} \)
7 \( 1 + 1.17e4T + 2.82e8T^{2} \)
11 \( 1 + (2.67e3 - 2.67e3i)T - 2.59e10iT^{2} \)
13 \( 1 + (4.43e5 - 4.43e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 1.39e4T + 2.01e12T^{2} \)
19 \( 1 + (-8.14e5 - 8.14e5i)T + 6.13e12iT^{2} \)
23 \( 1 + 4.54e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.78e5 + 1.78e5i)T - 4.20e14iT^{2} \)
31 \( 1 - 2.70e5iT - 8.19e14T^{2} \)
37 \( 1 + (4.38e7 + 4.38e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 4.64e6iT - 1.34e16T^{2} \)
43 \( 1 + (-8.47e6 + 8.47e6i)T - 2.16e16iT^{2} \)
47 \( 1 - 1.88e8iT - 5.25e16T^{2} \)
53 \( 1 + (-3.39e8 - 3.39e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (-3.20e7 + 3.20e7i)T - 5.11e17iT^{2} \)
61 \( 1 + (1.63e8 - 1.63e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (7.50e8 + 7.50e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.85e9T + 3.25e18T^{2} \)
73 \( 1 + 3.73e9iT - 4.29e18T^{2} \)
79 \( 1 - 5.16e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.21e9 + 3.21e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 6.63e9iT - 3.11e19T^{2} \)
97 \( 1 + 8.38e9T + 7.37e19T^{2} \)
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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

−16.69060169445668123524475745489, −16.10993806660039429678695076133, −15.21650142595467567816470440921, −13.99967944287562108643818285933, −12.37029679536536173639125647187, −9.670819116614235163562237441253, −8.845029491539486637266140983228, −7.63187668148562712859611183447, −4.75986411007563219750885597119, −3.81006650569123821362270332054, 0.22639384122912147967296146007, 2.58973535933531780210954057389, 3.42666947484000620611875648624, 7.17174960280626784001411096378, 8.204511992935502812498451026759, 10.09609863225443913643393688356, 11.83020177635537700607623203461, 12.86818147317411140588642241651, 14.21467536683557453547137001487, 15.21933154588855006201463752360

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