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Properties

Label 2-1664-16.5-c1-0-16
Degree $2$
Conductor $1664$
Sign $0.994 + 0.108i$
Analytic cond. $13.2871$
Root an. cond. $3.64514$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.140 + 0.140i)3-s + (−1.99 − 1.99i)5-s − 0.679i·7-s + 2.96i·9-s + (0.156 + 0.156i)11-s + (−0.707 + 0.707i)13-s + 0.561·15-s + 3.22·17-s + (−0.408 + 0.408i)19-s + (0.0954 + 0.0954i)21-s + 6.63i·23-s + 2.99i·25-s + (−0.837 − 0.837i)27-s + (4.96 − 4.96i)29-s − 2.26·31-s + ⋯
L(s)  = 1  + (−0.0811 + 0.0811i)3-s + (−0.894 − 0.894i)5-s − 0.256i·7-s + 0.986i·9-s + (0.0472 + 0.0472i)11-s + (−0.196 + 0.196i)13-s + 0.145·15-s + 0.783·17-s + (−0.0937 + 0.0937i)19-s + (0.0208 + 0.0208i)21-s + 1.38i·23-s + 0.598i·25-s + (−0.161 − 0.161i)27-s + (0.921 − 0.921i)29-s − 0.406·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(13.2871\)
Root analytic conductor: \(3.64514\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :1/2),\ 0.994 + 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318541819\)
\(L(\frac12)\) \(\approx\) \(1.318541819\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.140 - 0.140i)T - 3iT^{2} \)
5 \( 1 + (1.99 + 1.99i)T + 5iT^{2} \)
7 \( 1 + 0.679iT - 7T^{2} \)
11 \( 1 + (-0.156 - 0.156i)T + 11iT^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 + (0.408 - 0.408i)T - 19iT^{2} \)
23 \( 1 - 6.63iT - 23T^{2} \)
29 \( 1 + (-4.96 + 4.96i)T - 29iT^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + (2.37 + 2.37i)T + 37iT^{2} \)
41 \( 1 + 5.79iT - 41T^{2} \)
43 \( 1 + (-4.85 - 4.85i)T + 43iT^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + (-0.139 - 0.139i)T + 53iT^{2} \)
59 \( 1 + (0.930 + 0.930i)T + 59iT^{2} \)
61 \( 1 + (-1.62 + 1.62i)T - 61iT^{2} \)
67 \( 1 + (-4.82 + 4.82i)T - 67iT^{2} \)
71 \( 1 - 15.4iT - 71T^{2} \)
73 \( 1 + 9.54iT - 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + (-4.28 + 4.28i)T - 83iT^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 - 5.00T + 97T^{2} \)
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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

−9.264573089852518387067926247689, −8.489972866730689053429503987916, −7.67168450360657012065847790307, −7.35367131918563733061165299988, −5.92423407778891994375873344030, −5.14775183211871495977770960890, −4.36104504822654840974021136354, −3.60908915195346145330745795360, −2.19556488363346207082730168255, −0.825371492322229005148206401762, 0.78350242879338049243548444090, 2.57427037206689027573345477836, 3.38764535083166192084252268146, 4.17535670011059333890816053616, 5.34235158509972530409637336475, 6.35113317435479808253399195935, 6.95516802293275900547802022697, 7.69522142256883854210680111354, 8.580714598989056339112396601943, 9.286990941987280620364046968303

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