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Properties

Label 2-2e4-16.13-c11-0-13
Degree $2$
Conductor $16$
Sign $-0.946 + 0.323i$
Analytic cond. $12.2934$
Root an. cond. $3.50620$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−31.4 − 32.5i)2-s + (219. + 219. i)3-s + (−64.7 + 2.04e3i)4-s + (−4.57e3 + 4.57e3i)5-s + (221. − 1.40e4i)6-s − 2.77e3i·7-s + (6.85e4 − 6.23e4i)8-s − 8.09e4i·9-s + (2.92e5 + 4.62e3i)10-s + (1.73e4 − 1.73e4i)11-s + (−4.63e5 + 4.34e5i)12-s + (−1.16e6 − 1.16e6i)13-s + (−9.02e4 + 8.74e4i)14-s − 2.00e6·15-s + (−4.18e6 − 2.65e5i)16-s − 7.63e6·17-s + ⋯
L(s)  = 1  + (−0.695 − 0.718i)2-s + (0.521 + 0.521i)3-s + (−0.0316 + 0.999i)4-s + (−0.654 + 0.654i)5-s + (0.0116 − 0.736i)6-s − 0.0624i·7-s + (0.739 − 0.672i)8-s − 0.456i·9-s + (0.925 + 0.0146i)10-s + (0.0324 − 0.0324i)11-s + (−0.537 + 0.504i)12-s + (−0.871 − 0.871i)13-s + (−0.0448 + 0.0434i)14-s − 0.682·15-s + (−0.998 − 0.0631i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.946 + 0.323i$
Analytic conductor: \(12.2934\)
Root analytic conductor: \(3.50620\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :11/2),\ -0.946 + 0.323i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0465976 - 0.280306i\)
\(L(\frac12)\) \(\approx\) \(0.0465976 - 0.280306i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (31.4 + 32.5i)T \)
good3 \( 1 + (-219. - 219. i)T + 1.77e5iT^{2} \)
5 \( 1 + (4.57e3 - 4.57e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 2.77e3iT - 1.97e9T^{2} \)
11 \( 1 + (-1.73e4 + 1.73e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (1.16e6 + 1.16e6i)T + 1.79e12iT^{2} \)
17 \( 1 + 7.63e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.62e6 - 1.62e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 3.41e7iT - 9.52e14T^{2} \)
29 \( 1 + (5.22e7 + 5.22e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 1.70e8T + 2.54e16T^{2} \)
37 \( 1 + (-1.36e8 + 1.36e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 9.43e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.75e8 - 1.75e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 2.33e9T + 2.47e18T^{2} \)
53 \( 1 + (2.13e9 - 2.13e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (2.73e8 - 2.73e8i)T - 3.01e19iT^{2} \)
61 \( 1 + (-7.10e9 - 7.10e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-6.25e9 - 6.25e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 3.05e9iT - 2.31e20T^{2} \)
73 \( 1 - 3.14e10iT - 3.13e20T^{2} \)
79 \( 1 + 4.76e10T + 7.47e20T^{2} \)
83 \( 1 + (4.66e10 + 4.66e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 1.17e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.59e10T + 7.15e21T^{2} \)
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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

−15.77327808550854995755803168298, −14.67991901821559545496927679765, −12.75684912968735280032103433801, −11.30856329681904076687871976439, −10.08000265637749388069638107553, −8.718472188706325633438695289158, −7.21842202348919532624457332449, −4.00694129891866700580428329156, −2.67818318629679318960022481324, −0.13994910323790668179551086072, 1.82800152269200506951571052150, 4.81930990734606412681323476327, 7.00903758893142444840852918991, 8.167095316334171049672989656461, 9.357048938008969833879812206935, 11.34637072883661459107796045496, 13.16477836835440327236578235931, 14.50780505049124955957988302537, 15.86009144698599562008986843159, 16.89129593974032714749042196307

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