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Properties

Label 2-2664-2664.2515-c0-0-9
Degree $2$
Conductor $2664$
Sign $-0.997 + 0.0697i$
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 + (0.309 − 0.951i)3-s + (−0.499 + 0.866i)4-s + (0.809 − 1.40i)5-s + (−0.978 + 0.207i)6-s + 0.999·8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (0.104 + 0.181i)11-s + (0.669 + 0.743i)12-s + (0.978 − 1.69i)13-s + (−1.08 − 1.20i)15-s + (−0.5 − 0.866i)16-s + (−0.104 + 0.994i)18-s + (0.809 + 1.40i)20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.309 − 0.951i)3-s + (−0.499 + 0.866i)4-s + (0.809 − 1.40i)5-s + (−0.978 + 0.207i)6-s + 0.999·8-s + (−0.809 − 0.587i)9-s − 1.61·10-s + (0.104 + 0.181i)11-s + (0.669 + 0.743i)12-s + (0.978 − 1.69i)13-s + (−1.08 − 1.20i)15-s + (−0.5 − 0.866i)16-s + (−0.104 + 0.994i)18-s + (0.809 + 1.40i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.997 + 0.0697i$
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: \((2,\ 2664,\ (\ :0),\ -0.997 + 0.0697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170083327\)
\(L(\frac12)\) \(\approx\) \(1.170083327\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 - T \)
good5 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.104 - 0.181i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.913 - 1.58i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.209T + T^{2} \)
79 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.627700275435269704440780518661, −8.136471636763280044351723078232, −7.59245195072904525938047859753, −6.26780936476822113280540416313, −5.59775929368079165496528381608, −4.70088214682091588317923140610, −3.51776370803522277985800104413, −2.65115319203962331133573966535, −1.49997070978960279014676170433, −0.966974440801925629202920328063, 1.86833849006513229208962784990, 2.84291476468843045361797521246, 4.04176662784999960039386086690, 4.66940917096069291024120852549, 5.96222906089548492739250222347, 6.30895690681838775960257796523, 6.92881169641038235446270690396, 8.045565590114688624546582257866, 8.720352235270851815464179804329, 9.350359165837287918514765247327

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