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Properties

Label 2-2e4-16.3-c10-0-4
Degree $2$
Conductor $16$
Sign $-0.401 - 0.915i$
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  + (−31.9 + 0.164i)2-s + (−221. + 221. i)3-s + (1.02e3 − 10.5i)4-s + (2.57e3 − 2.57e3i)5-s + (7.06e3 − 7.13e3i)6-s + 4.01e3·7-s + (−3.27e4 + 504. i)8-s − 3.95e4i·9-s + (−8.21e4 + 8.29e4i)10-s + (1.38e5 + 1.38e5i)11-s + (−2.24e5 + 2.29e5i)12-s + (−1.81e4 − 1.81e4i)13-s + (−1.28e5 + 659. i)14-s + 1.14e6i·15-s + (1.04e6 − 2.15e4i)16-s − 1.82e6·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00512i)2-s + (−0.913 + 0.913i)3-s + (0.999 − 0.0102i)4-s + (0.825 − 0.825i)5-s + (0.908 − 0.918i)6-s + 0.239·7-s + (−0.999 + 0.0153i)8-s − 0.669i·9-s + (−0.821 + 0.829i)10-s + (0.857 + 0.857i)11-s + (−0.904 + 0.922i)12-s + (−0.0487 − 0.0487i)13-s + (−0.239 + 0.00122i)14-s + 1.50i·15-s + (0.999 − 0.0205i)16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.401 - 0.915i$
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: \((2,\ 16,\ (\ :5),\ -0.401 - 0.915i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.399322 + 0.611107i\)
\(L(\frac12)\) \(\approx\) \(0.399322 + 0.611107i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (31.9 - 0.164i)T \)
good3 \( 1 + (221. - 221. i)T - 5.90e4iT^{2} \)
5 \( 1 + (-2.57e3 + 2.57e3i)T - 9.76e6iT^{2} \)
7 \( 1 - 4.01e3T + 2.82e8T^{2} \)
11 \( 1 + (-1.38e5 - 1.38e5i)T + 2.59e10iT^{2} \)
13 \( 1 + (1.81e4 + 1.81e4i)T + 1.37e11iT^{2} \)
17 \( 1 + 1.82e6T + 2.01e12T^{2} \)
19 \( 1 + (2.84e6 - 2.84e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 7.52e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.29e7 - 1.29e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 3.98e7iT - 8.19e14T^{2} \)
37 \( 1 + (-1.07e7 + 1.07e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 2.03e8iT - 1.34e16T^{2} \)
43 \( 1 + (5.22e7 + 5.22e7i)T + 2.16e16iT^{2} \)
47 \( 1 - 2.90e8iT - 5.25e16T^{2} \)
53 \( 1 + (-2.64e8 + 2.64e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-6.22e8 - 6.22e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (1.14e9 + 1.14e9i)T + 7.13e17iT^{2} \)
67 \( 1 + (2.36e8 - 2.36e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 4.62e8T + 3.25e18T^{2} \)
73 \( 1 - 1.05e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.84e8iT - 9.46e18T^{2} \)
83 \( 1 + (-1.28e9 + 1.28e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 4.28e9iT - 3.11e19T^{2} \)
97 \( 1 - 5.82e9T + 7.37e19T^{2} \)
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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

−17.13379170201633621571115269272, −16.30654749863756528417051003034, −14.90809226242462315375506232368, −12.55979188940808678980029104501, −11.08734023638726558277041254433, −9.928109252196330124876629068564, −8.806858404365842678008682628976, −6.43299462169764762714920042286, −4.79255693430345726159032248493, −1.54824512451781451248311875214, 0.53807406042795822323155157335, 2.21424690221883399102230421996, 6.19248530111209933957598717755, 6.88023468136317692853047450617, 8.945201502297879253910421837110, 10.77467497120731318823177236137, 11.58945810282814314645272665261, 13.34656014693760942263199323035, 15.08864042046308531501303262949, 17.02437190225032278558589156197

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