Properties

Label 2-2e4-16.3-c10-0-15
Degree $2$
Conductor $16$
Sign $0.978 + 0.205i$
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  + (27.4 + 16.5i)2-s + (305. − 305. i)3-s + (477. + 905. i)4-s + (155. − 155. i)5-s + (1.34e4 − 3.32e3i)6-s + 8.52e3·7-s + (−1.86e3 + 3.27e4i)8-s − 1.28e5i·9-s + (6.82e3 − 1.68e3i)10-s + (−1.64e5 − 1.64e5i)11-s + (4.23e5 + 1.30e5i)12-s + (4.09e5 + 4.09e5i)13-s + (2.33e5 + 1.40e5i)14-s − 9.50e4i·15-s + (−5.91e5 + 8.65e5i)16-s + 9.41e5·17-s + ⋯
L(s)  = 1  + (0.856 + 0.516i)2-s + (1.25 − 1.25i)3-s + (0.466 + 0.884i)4-s + (0.0496 − 0.0496i)5-s + (1.72 − 0.428i)6-s + 0.507·7-s + (−0.0569 + 0.998i)8-s − 2.17i·9-s + (0.0682 − 0.0168i)10-s + (−1.02 − 1.02i)11-s + (1.70 + 0.525i)12-s + (1.10 + 1.10i)13-s + (0.434 + 0.261i)14-s − 0.125i·15-s + (−0.564 + 0.825i)16-s + 0.663·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.978 + 0.205i$
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: \((2,\ 16,\ (\ :5),\ 0.978 + 0.205i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.06788 - 0.422178i\)
\(L(\frac12)\) \(\approx\) \(4.06788 - 0.422178i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-27.4 - 16.5i)T \)
good3 \( 1 + (-305. + 305. i)T - 5.90e4iT^{2} \)
5 \( 1 + (-155. + 155. i)T - 9.76e6iT^{2} \)
7 \( 1 - 8.52e3T + 2.82e8T^{2} \)
11 \( 1 + (1.64e5 + 1.64e5i)T + 2.59e10iT^{2} \)
13 \( 1 + (-4.09e5 - 4.09e5i)T + 1.37e11iT^{2} \)
17 \( 1 - 9.41e5T + 2.01e12T^{2} \)
19 \( 1 + (1.22e6 - 1.22e6i)T - 6.13e12iT^{2} \)
23 \( 1 + 7.27e6T + 4.14e13T^{2} \)
29 \( 1 + (1.86e7 + 1.86e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 1.16e7iT - 8.19e14T^{2} \)
37 \( 1 + (-1.73e7 + 1.73e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.55e8iT - 1.34e16T^{2} \)
43 \( 1 + (9.07e7 + 9.07e7i)T + 2.16e16iT^{2} \)
47 \( 1 - 1.93e7iT - 5.25e16T^{2} \)
53 \( 1 + (-1.80e8 + 1.80e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-2.64e8 - 2.64e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-7.17e8 - 7.17e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (1.41e8 - 1.41e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 1.78e9T + 3.25e18T^{2} \)
73 \( 1 + 2.36e9iT - 4.29e18T^{2} \)
79 \( 1 + 5.63e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.51e9 + 3.51e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 1.13e9iT - 3.11e19T^{2} \)
97 \( 1 + 2.05e9T + 7.37e19T^{2} \)
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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

−16.35794445609461599415362324246, −14.80800946274813697888568916463, −13.79358381299879950513184713072, −13.12219662264109486936037506689, −11.59348195280189957747349393578, −8.559259390239735605498558493328, −7.70606203550562283282835506577, −6.09077195656071968555358694216, −3.52067319590515954594423210863, −1.87362916896809447843539493909, 2.32745914643773447599067825668, 3.77630175350256209032653335075, 5.16650304672537250116860148943, 8.099284369780248877282979142518, 9.904892062482355876479301109550, 10.77834521755343135831037764531, 12.91928396339760076130191959738, 14.18101735812016418570196204202, 15.18305006493653883017393482559, 15.91662638686314997835281600996

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