Properties

Label 2-2664-2664.2515-c0-0-6
Degree $2$
Conductor $2664$
Sign $0.848 + 0.529i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
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  + (0.5 + 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.499 + 0.866i)4-s + (0.309 − 0.535i)5-s + (0.104 − 0.994i)6-s − 0.999·8-s + (0.309 + 0.951i)9-s + 0.618·10-s + (−0.669 − 1.15i)11-s + (0.913 − 0.406i)12-s + (−0.104 + 0.181i)13-s + (−0.564 + 0.251i)15-s + (−0.5 − 0.866i)16-s + (−0.669 + 0.743i)18-s + (0.309 + 0.535i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.499 + 0.866i)4-s + (0.309 − 0.535i)5-s + (0.104 − 0.994i)6-s − 0.999·8-s + (0.309 + 0.951i)9-s + 0.618·10-s + (−0.669 − 1.15i)11-s + (0.913 − 0.406i)12-s + (−0.104 + 0.181i)13-s + (−0.564 + 0.251i)15-s + (−0.5 − 0.866i)16-s + (−0.669 + 0.743i)18-s + (0.309 + 0.535i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (2515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9748772351\)
\(L(\frac12)\) \(\approx\) \(0.9748772351\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + T \)
good5 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.669 + 1.15i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.104 - 0.181i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.913 + 1.58i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.104 + 0.181i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.978 + 1.69i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.33T + T^{2} \)
79 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−8.711004557221679239983108117324, −8.066415582333957350191083536837, −7.33456324858770289695469786567, −6.47831838829440436704150021400, −5.96575781046311183078218785527, −5.12511763461191213133794369065, −4.72449770050354488101958759899, −3.43762923178478532971699423892, −2.31120160908014076597295096673, −0.62985556920797959521104501074, 1.37809701761524206132898157254, 2.61435072986306681337286900634, 3.44033045037640322299623649102, 4.45327425892102002172771013666, 5.10000914200769290649881939433, 5.70793966813640636419896794149, 6.65746958700625210386441604091, 7.32273988406750788877307680001, 8.644574690554241278461109553553, 9.582217395222222699501071346745

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