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Properties

Label 2-1664-104.45-c0-0-0
Degree $2$
Conductor $1664$
Sign $0.999 + 0.0386i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s − 0.732i·25-s + (0.866 + 0.5i)29-s + (0.5 + 1.86i)37-s + (−0.133 − 0.5i)41-s + (0.133 − 0.5i)45-s + (0.866 + 0.5i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (0.133 + 0.5i)65-s + (−0.366 − 0.366i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.366 + 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s − 0.732i·25-s + (0.866 + 0.5i)29-s + (0.5 + 1.86i)37-s + (−0.133 − 0.5i)41-s + (0.133 − 0.5i)45-s + (0.866 + 0.5i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (0.133 + 0.5i)65-s + (−0.366 − 0.366i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.999 + 0.0386i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ 0.999 + 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227507381\)
\(L(\frac12)\) \(\approx\) \(1.227507381\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.578110626071659466503419977348, −8.758127735280383299150085853967, −8.116185122779472749510297790046, −6.95822620144569738246858995782, −6.36101149523914946322276238178, −5.65263238595692111494221067955, −4.55242880869682736324416239262, −3.47919866783554909880937613180, −2.71118838504177062703637670153, −1.23165565578250541940892833199, 1.31649520234461198368297876567, 2.54662471610221815395169940713, 3.60154267394979434876250960383, 4.68035977380746878031591787551, 5.66212227067309262144641469101, 6.01603026792887291513676952986, 7.35620770158059976461434876102, 8.024382942943451621202737203697, 8.731288592940187410922561649904, 9.493567834588040111510036133535

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