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Properties

Label 2-1664-16.5-c1-0-26
Degree $2$
Conductor $1664$
Sign $0.651 + 0.758i$
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  + (1.33 − 1.33i)3-s + (−1.78 − 1.78i)5-s + 4.71i·7-s − 0.544i·9-s + (−1.20 − 1.20i)11-s + (0.707 − 0.707i)13-s − 4.75·15-s + 5.21·17-s + (0.338 − 0.338i)19-s + (6.28 + 6.28i)21-s − 6.09i·23-s + 1.38i·25-s + (3.26 + 3.26i)27-s + (4.52 − 4.52i)29-s + 5.92·31-s + ⋯
L(s)  = 1  + (0.768 − 0.768i)3-s + (−0.799 − 0.799i)5-s + 1.78i·7-s − 0.181i·9-s + (−0.363 − 0.363i)11-s + (0.196 − 0.196i)13-s − 1.22·15-s + 1.26·17-s + (0.0776 − 0.0776i)19-s + (1.37 + 1.37i)21-s − 1.27i·23-s + 0.277i·25-s + (0.629 + 0.629i)27-s + (0.841 − 0.841i)29-s + 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.651 + 0.758i$
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: \((2,\ 1664,\ (\ :1/2),\ 0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.972195849\)
\(L(\frac12)\) \(\approx\) \(1.972195849\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 1.33i)T - 3iT^{2} \)
5 \( 1 + (1.78 + 1.78i)T + 5iT^{2} \)
7 \( 1 - 4.71iT - 7T^{2} \)
11 \( 1 + (1.20 + 1.20i)T + 11iT^{2} \)
17 \( 1 - 5.21T + 17T^{2} \)
19 \( 1 + (-0.338 + 0.338i)T - 19iT^{2} \)
23 \( 1 + 6.09iT - 23T^{2} \)
29 \( 1 + (-4.52 + 4.52i)T - 29iT^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + (3.58 + 3.58i)T + 37iT^{2} \)
41 \( 1 + 6.11iT - 41T^{2} \)
43 \( 1 + (-0.646 - 0.646i)T + 43iT^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (-5.55 - 5.55i)T + 53iT^{2} \)
59 \( 1 + (-6.12 - 6.12i)T + 59iT^{2} \)
61 \( 1 + (-3.67 + 3.67i)T - 61iT^{2} \)
67 \( 1 + (-2.28 + 2.28i)T - 67iT^{2} \)
71 \( 1 - 8.53iT - 71T^{2} \)
73 \( 1 - 2.44iT - 73T^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 + (2.65 - 2.65i)T - 83iT^{2} \)
89 \( 1 - 7.86iT - 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.735002294084724175695926676090, −8.483342780173880853911448163486, −8.019646988921992460909870760357, −7.05762272368757641553344069323, −5.88051143782167681635698683030, −5.29377297046216543171492475461, −4.19708580089818088634023300316, −2.88037080527094187097631257697, −2.35638319609643736010472327395, −0.878662394891597460412171199042, 1.08901273816183986551064100793, 2.96605219845439213976280084404, 3.59032278200537603881952671878, 4.10919169121256167950850483050, 5.09345329287903282134257035490, 6.54751427197522739076505893958, 7.29211196930123805932138082069, 7.77806627999514791584911795806, 8.591243846692918022222662172251, 9.804379067219855594629242337451

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