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Properties

Label 2-2e4-16.13-c9-0-4
Degree $2$
Conductor $16$
Sign $-0.965 + 0.259i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.735 + 22.6i)2-s + (100. + 100. i)3-s + (−510. − 33.2i)4-s + (−601. + 601. i)5-s + (−2.33e3 + 2.18e3i)6-s + 5.43e3i·7-s + (1.12e3 − 1.15e4i)8-s + 340. i·9-s + (−1.31e4 − 1.40e4i)10-s + (−2.28e4 + 2.28e4i)11-s + (−4.77e4 − 5.44e4i)12-s + (−7.39e4 − 7.39e4i)13-s + (−1.22e5 − 3.99e3i)14-s − 1.20e5·15-s + (2.59e5 + 3.39e4i)16-s − 1.09e5·17-s + ⋯
L(s)  = 1  + (−0.0324 + 0.999i)2-s + (0.713 + 0.713i)3-s + (−0.997 − 0.0649i)4-s + (−0.430 + 0.430i)5-s + (−0.735 + 0.689i)6-s + 0.856i·7-s + (0.0973 − 0.995i)8-s + 0.0172i·9-s + (−0.416 − 0.444i)10-s + (−0.470 + 0.470i)11-s + (−0.665 − 0.758i)12-s + (−0.718 − 0.718i)13-s + (−0.855 − 0.0278i)14-s − 0.613·15-s + (0.991 + 0.129i)16-s − 0.316·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ -0.965 + 0.259i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.170548 - 1.29075i\)
\(L(\frac12)\) \(\approx\) \(0.170548 - 1.29075i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.735 - 22.6i)T \)
good3 \( 1 + (-100. - 100. i)T + 1.96e4iT^{2} \)
5 \( 1 + (601. - 601. i)T - 1.95e6iT^{2} \)
7 \( 1 - 5.43e3iT - 4.03e7T^{2} \)
11 \( 1 + (2.28e4 - 2.28e4i)T - 2.35e9iT^{2} \)
13 \( 1 + (7.39e4 + 7.39e4i)T + 1.06e10iT^{2} \)
17 \( 1 + 1.09e5T + 1.18e11T^{2} \)
19 \( 1 + (-4.22e5 - 4.22e5i)T + 3.22e11iT^{2} \)
23 \( 1 - 1.59e6iT - 1.80e12T^{2} \)
29 \( 1 + (-4.82e6 - 4.82e6i)T + 1.45e13iT^{2} \)
31 \( 1 + 5.33e6T + 2.64e13T^{2} \)
37 \( 1 + (1.16e7 - 1.16e7i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.31e6iT - 3.27e14T^{2} \)
43 \( 1 + (-2.23e7 + 2.23e7i)T - 5.02e14iT^{2} \)
47 \( 1 - 2.41e7T + 1.11e15T^{2} \)
53 \( 1 + (-4.65e6 + 4.65e6i)T - 3.29e15iT^{2} \)
59 \( 1 + (-8.29e7 + 8.29e7i)T - 8.66e15iT^{2} \)
61 \( 1 + (7.56e7 + 7.56e7i)T + 1.16e16iT^{2} \)
67 \( 1 + (1.25e8 + 1.25e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.00e8iT - 4.58e16T^{2} \)
73 \( 1 - 4.42e8iT - 5.88e16T^{2} \)
79 \( 1 + 6.19e7T + 1.19e17T^{2} \)
83 \( 1 + (-1.21e8 - 1.21e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 1.01e9iT - 3.50e17T^{2} \)
97 \( 1 - 7.73e8T + 7.60e17T^{2} \)
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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

−17.58454948552892225788101411531, −15.73752709791133182270873904496, −15.27335624089305122004711131753, −14.21191328450419800906287994150, −12.39937189259573142635786225643, −10.06185287998194268172943293161, −8.829783884020297302254927843318, −7.38701096830504547785461598133, −5.28029949140139237826320302671, −3.35721443323804054356674116232, 0.63086143129584134518820503787, 2.50138250542731930311308354254, 4.46430118515178607547812215933, 7.54803807376743763640417622284, 8.846922543545154080237106914695, 10.60547386868734618616161593855, 12.15357513270349579575790567380, 13.40006729892255408093908030736, 14.21613427776931073070291382900, 16.42375923574457068703402732635

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