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Properties

Label 2-2e4-16.11-c10-0-3
Degree $2$
Conductor $16$
Sign $-0.675 - 0.737i$
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  + (28.0 + 15.3i)2-s + (−317. − 317. i)3-s + (550. + 863. i)4-s + (1.56e3 + 1.56e3i)5-s + (−4.01e3 − 1.37e4i)6-s − 1.54e4·7-s + (2.14e3 + 3.26e4i)8-s + 1.42e5i·9-s + (1.98e4 + 6.79e4i)10-s + (−1.41e5 + 1.41e5i)11-s + (9.94e4 − 4.48e5i)12-s + (−2.15e5 + 2.15e5i)13-s + (−4.34e5 − 2.38e5i)14-s − 9.92e5i·15-s + (−4.42e5 + 9.50e5i)16-s + 1.64e6·17-s + ⋯
L(s)  = 1  + (0.876 + 0.480i)2-s + (−1.30 − 1.30i)3-s + (0.537 + 0.843i)4-s + (0.500 + 0.500i)5-s + (−0.516 − 1.77i)6-s − 0.922·7-s + (0.0656 + 0.997i)8-s + 2.41i·9-s + (0.198 + 0.679i)10-s + (−0.880 + 0.880i)11-s + (0.399 − 1.80i)12-s + (−0.580 + 0.580i)13-s + (−0.808 − 0.443i)14-s − 1.30i·15-s + (−0.422 + 0.906i)16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.675 - 0.737i$
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: \((2,\ 16,\ (\ :5),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.367486 + 0.835492i\)
\(L(\frac12)\) \(\approx\) \(0.367486 + 0.835492i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-28.0 - 15.3i)T \)
good3 \( 1 + (317. + 317. i)T + 5.90e4iT^{2} \)
5 \( 1 + (-1.56e3 - 1.56e3i)T + 9.76e6iT^{2} \)
7 \( 1 + 1.54e4T + 2.82e8T^{2} \)
11 \( 1 + (1.41e5 - 1.41e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (2.15e5 - 2.15e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 1.64e6T + 2.01e12T^{2} \)
19 \( 1 + (1.70e6 + 1.70e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 1.04e6T + 4.14e13T^{2} \)
29 \( 1 + (7.08e6 - 7.08e6i)T - 4.20e14iT^{2} \)
31 \( 1 - 3.47e7iT - 8.19e14T^{2} \)
37 \( 1 + (9.40e7 + 9.40e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 4.61e7iT - 1.34e16T^{2} \)
43 \( 1 + (-6.50e7 + 6.50e7i)T - 2.16e16iT^{2} \)
47 \( 1 - 1.86e8iT - 5.25e16T^{2} \)
53 \( 1 + (-3.25e8 - 3.25e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (-4.67e8 + 4.67e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (2.61e8 - 2.61e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (8.34e7 + 8.34e7i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.30e9T + 3.25e18T^{2} \)
73 \( 1 - 1.11e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.68e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.57e9 - 3.57e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 4.11e8iT - 3.11e19T^{2} \)
97 \( 1 - 1.30e10T + 7.37e19T^{2} \)
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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

−17.16209942746949101089134121790, −16.02596703201069583830512796065, −14.13993762834636011799373008560, −12.83965424860817350019516313283, −12.23105206679432524163573906918, −10.56533165624941228078866102733, −7.36577977796344853701310465939, −6.52765111683995899644739798012, −5.25387588739700183915791028147, −2.29689339425807676890953648091, 0.35233183313251509534447657950, 3.48536185971143799024683943678, 5.22132232779823694469795971247, 5.98601907850716465793172299734, 9.775418775877709848686904529458, 10.48787456641375274174969462384, 11.96583671876085335491627417524, 13.13452920722677812461303849349, 15.05851835194770224840165887563, 16.17804970063858480720486562762

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