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Properties

Label 2-2e4-16.3-c10-0-12
Degree $2$
Conductor $16$
Sign $-0.872 + 0.488i$
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  + (−29.1 + 13.1i)2-s + (32.7 − 32.7i)3-s + (680. − 765. i)4-s + (−1.64e3 + 1.64e3i)5-s + (−526. + 1.38e3i)6-s + 1.66e4·7-s + (−9.81e3 + 3.12e4i)8-s + 5.69e4i·9-s + (2.64e4 − 6.95e4i)10-s + (−2.06e5 − 2.06e5i)11-s + (−2.79e3 − 4.73e4i)12-s + (−2.60e5 − 2.60e5i)13-s + (−4.85e5 + 2.18e5i)14-s + 1.07e5i·15-s + (−1.23e5 − 1.04e6i)16-s − 1.45e6·17-s + ⋯
L(s)  = 1  + (−0.912 + 0.409i)2-s + (0.134 − 0.134i)3-s + (0.664 − 0.747i)4-s + (−0.526 + 0.526i)5-s + (−0.0676 + 0.178i)6-s + 0.990·7-s + (−0.299 + 0.954i)8-s + 0.963i·9-s + (0.264 − 0.695i)10-s + (−1.28 − 1.28i)11-s + (−0.0112 − 0.190i)12-s + (−0.702 − 0.702i)13-s + (−0.903 + 0.405i)14-s + 0.141i·15-s + (−0.117 − 0.993i)16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0101939 - 0.0390620i\)
\(L(\frac12)\) \(\approx\) \(0.0101939 - 0.0390620i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (29.1 - 13.1i)T \)
good3 \( 1 + (-32.7 + 32.7i)T - 5.90e4iT^{2} \)
5 \( 1 + (1.64e3 - 1.64e3i)T - 9.76e6iT^{2} \)
7 \( 1 - 1.66e4T + 2.82e8T^{2} \)
11 \( 1 + (2.06e5 + 2.06e5i)T + 2.59e10iT^{2} \)
13 \( 1 + (2.60e5 + 2.60e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 1.45e6T + 2.01e12T^{2} \)
19 \( 1 + (-3.82e4 + 3.82e4i)T - 6.13e12iT^{2} \)
23 \( 1 + 7.64e6T + 4.14e13T^{2} \)
29 \( 1 + (1.59e7 + 1.59e7i)T + 4.20e14iT^{2} \)
31 \( 1 + 1.35e6iT - 8.19e14T^{2} \)
37 \( 1 + (3.25e7 - 3.25e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.46e8iT - 1.34e16T^{2} \)
43 \( 1 + (-1.13e7 - 1.13e7i)T + 2.16e16iT^{2} \)
47 \( 1 - 2.26e8iT - 5.25e16T^{2} \)
53 \( 1 + (-3.68e8 + 3.68e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-8.83e8 - 8.83e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (3.03e8 + 3.03e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (-8.18e8 + 8.18e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 2.31e9T + 3.25e18T^{2} \)
73 \( 1 + 1.65e9iT - 4.29e18T^{2} \)
79 \( 1 - 2.94e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.13e9 - 1.13e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 8.46e9iT - 3.11e19T^{2} \)
97 \( 1 - 9.19e9T + 7.37e19T^{2} \)
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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

−16.12326707575834564378365535604, −15.08699074448428050067228722305, −13.68134487018683953373755795725, −11.31809281856002896317768193704, −10.51344082801236544593762170508, −8.296096796799585050854065685979, −7.57400604406957007324353526471, −5.40004410341366941282328664617, −2.35153341888143353594315142954, −0.02311235259596096178484817760, 2.03874748188586258103533317677, 4.41313285323988316049398067799, 7.29520535922268120041004963998, 8.589596929134800414514020450423, 10.00478941941096730589117976539, 11.61488863464844031582223904246, 12.62915699294517459452325891883, 14.91463462994695082133266898181, 16.01429878122684330076295876364, 17.58487960369885552085882630430

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