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Properties

Label 2-2664-296.147-c0-0-3
Degree $2$
Conductor $2664$
Sign $0.382 + 0.923i$
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  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s − 1.84·5-s + 1.41i·7-s + (−0.923 + 0.382i)8-s + (−0.707 + 1.70i)10-s + (1.30 + 0.541i)14-s + i·16-s − 1.84i·17-s + (1.30 + 1.30i)20-s + 1.84·23-s + 2.41·25-s + (1.00 − i)28-s + 0.765·29-s + (0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s − 1.84·5-s + 1.41i·7-s + (−0.923 + 0.382i)8-s + (−0.707 + 1.70i)10-s + (1.30 + 0.541i)14-s + i·16-s − 1.84i·17-s + (1.30 + 1.30i)20-s + 1.84·23-s + 2.41·25-s + (1.00 − i)28-s + 0.765·29-s + (0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8998823818\)
\(L(\frac12)\) \(\approx\) \(0.8998823818\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
3 \( 1 \)
37 \( 1 + iT \)
good5 \( 1 + 1.84T + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.84iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.84T + T^{2} \)
29 \( 1 - 0.765T + T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.84iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 0.765iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.977106575892864554128796470041, −8.377680652823940299448848468934, −7.43168210038482305837555609867, −6.65024640637757604134622637285, −5.32804851592166107262293779607, −4.93571787195976445387616057831, −4.01637560996657393869427880899, −3.03086757903055465057292838216, −2.59298913796615371330345718112, −0.795240026356658582282098779273, 0.874008472681171131240413780436, 3.24879307978255272710451852445, 3.73357428342909657581953319407, 4.42916645294775167487959775390, 5.04101526718071522720207282624, 6.47996042471081427569097084955, 6.91517655594228027672360788495, 7.64447584027773166729967179063, 8.203909973594439901147148697282, 8.677529477118951821468296493547

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