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Properties

Label 2-1664-104.43-c0-0-1
Degree $2$
Conductor $1664$
Sign $0.859 - 0.511i$
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  + 5-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−1.5 − 0.866i)29-s + (0.5 − 0.866i)37-s + (1.5 + 0.866i)41-s + (0.5 + 0.866i)45-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (−0.5 + 0.866i)65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯
L(s)  = 1  + 5-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−1.5 − 0.866i)29-s + (0.5 − 0.866i)37-s + (1.5 + 0.866i)41-s + (0.5 + 0.866i)45-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (−0.5 + 0.866i)65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + (0.5 + 0.866i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346736925\)
\(L(\frac12)\) \(\approx\) \(1.346736925\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.656862773094783729822313193724, −9.034741249827294693622439586181, −7.896352445748232486783599912200, −7.35190877051522795413768325868, −6.25473910676379170535148911962, −5.65844999543426001318701984712, −4.70021495154785196383616852738, −3.82009092850780163204768338587, −2.32973635006609149747485075238, −1.72031026303898862251822839286, 1.17912227529381637020140148377, 2.48208325063897938490477730072, 3.43122077937988360517335628199, 4.58586846033311780706815246636, 5.58746450045159077253227273293, 6.08566729852384318533631423164, 7.19665121896137471609823355336, 7.70230823770246258954333676380, 9.038409620286052314452912493558, 9.446518673043814835086443123822

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