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Properties

Label 2-2e4-16.3-c10-0-9
Degree $2$
Conductor $16$
Sign $-0.153 + 0.988i$
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  + (−19.3 − 25.5i)2-s + (−196. + 196. i)3-s + (−276. + 985. i)4-s + (−1.31e3 + 1.31e3i)5-s + (8.79e3 + 1.21e3i)6-s − 1.12e3·7-s + (3.04e4 − 1.20e4i)8-s − 1.79e4i·9-s + (5.87e4 + 8.09e3i)10-s + (−7.02e4 − 7.02e4i)11-s + (−1.39e5 − 2.47e5i)12-s + (−2.29e5 − 2.29e5i)13-s + (2.16e4 + 2.85e4i)14-s − 5.14e5i·15-s + (−8.95e5 − 5.45e5i)16-s + 3.84e4·17-s + ⋯
L(s)  = 1  + (−0.604 − 0.796i)2-s + (−0.807 + 0.807i)3-s + (−0.270 + 0.962i)4-s + (−0.419 + 0.419i)5-s + (1.13 + 0.155i)6-s − 0.0666·7-s + (0.930 − 0.366i)8-s − 0.304i·9-s + (0.587 + 0.0809i)10-s + (−0.436 − 0.436i)11-s + (−0.559 − 0.995i)12-s + (−0.617 − 0.617i)13-s + (0.0402 + 0.0531i)14-s − 0.677i·15-s + (−0.854 − 0.520i)16-s + 0.0270·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.153 + 0.988i$
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: \((2,\ 16,\ (\ :5),\ -0.153 + 0.988i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.286755 - 0.334811i\)
\(L(\frac12)\) \(\approx\) \(0.286755 - 0.334811i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (19.3 + 25.5i)T \)
good3 \( 1 + (196. - 196. i)T - 5.90e4iT^{2} \)
5 \( 1 + (1.31e3 - 1.31e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 1.12e3T + 2.82e8T^{2} \)
11 \( 1 + (7.02e4 + 7.02e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (2.29e5 + 2.29e5i)T + 1.37e11iT^{2} \)
17 \( 1 - 3.84e4T + 2.01e12T^{2} \)
19 \( 1 + (-2.95e6 + 2.95e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 8.89e6T + 4.14e13T^{2} \)
29 \( 1 + (5.33e6 + 5.33e6i)T + 4.20e14iT^{2} \)
31 \( 1 + 5.79e6iT - 8.19e14T^{2} \)
37 \( 1 + (-8.06e7 + 8.06e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.06e8iT - 1.34e16T^{2} \)
43 \( 1 + (1.52e8 + 1.52e8i)T + 2.16e16iT^{2} \)
47 \( 1 + 1.83e8iT - 5.25e16T^{2} \)
53 \( 1 + (5.33e8 - 5.33e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (1.89e8 + 1.89e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-4.05e8 - 4.05e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (-1.23e9 + 1.23e9i)T - 1.82e18iT^{2} \)
71 \( 1 + 1.29e9T + 3.25e18T^{2} \)
73 \( 1 + 2.45e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.29e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.53e9 + 1.53e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 5.49e9iT - 3.11e19T^{2} \)
97 \( 1 - 4.01e9T + 7.37e19T^{2} \)
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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

−16.54351566846849036400633283071, −15.36548531127702995575832453244, −13.20045973683929526353371038957, −11.52053676389559886573475384432, −10.78995425323497932003468370487, −9.477952607710513829022457635828, −7.53779253983728465363357502464, −4.98468297830885332453414208337, −3.09122255396830148050872528228, −0.32790617673192824230468507700, 1.19429973013059190632866592530, 5.08051768750062954870640599452, 6.66519548026067144296074676312, 7.87846685332320463492816281635, 9.683507100807039316681609077655, 11.52346238669777046036827599198, 12.84609890280595314044008291997, 14.57550100364563406589486254459, 16.09175103336858349102810356590, 17.06521249575160241672364636255

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