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Properties

Label 2-1536-24.5-c0-0-4
Degree $2$
Conductor $1536$
Sign $1$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·21-s + 1.00·25-s − 27-s − 1.41·29-s − 1.41·31-s + 2.00·35-s + 1.41·45-s + 1.00·49-s − 1.41·53-s + 2·59-s + 1.41·63-s − 1.00·75-s + 1.41·79-s + 81-s + 1.41·87-s + 1.41·93-s − 2·97-s + 1.41·101-s − 1.41·103-s − 2.00·105-s + 2·107-s + ⋯
L(s)  = 1  − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·21-s + 1.00·25-s − 27-s − 1.41·29-s − 1.41·31-s + 2.00·35-s + 1.41·45-s + 1.00·49-s − 1.41·53-s + 2·59-s + 1.41·63-s − 1.00·75-s + 1.41·79-s + 81-s + 1.41·87-s + 1.41·93-s − 2·97-s + 1.41·101-s − 1.41·103-s − 2.00·105-s + 2·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (257, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.200700432\)
\(L(\frac12)\) \(\approx\) \(1.200700432\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.704698545769151316221049114939, −9.061440982478982806773319922482, −7.949432902522207787558221081954, −7.14130516324120243245311158864, −6.21057481736321247637258704569, −5.42043489559398996493544921206, −5.05463650272466308974538619838, −3.91364828021056951322200487157, −2.13596525330082344886500652491, −1.46055280576320397413963830171, 1.46055280576320397413963830171, 2.13596525330082344886500652491, 3.91364828021056951322200487157, 5.05463650272466308974538619838, 5.42043489559398996493544921206, 6.21057481736321247637258704569, 7.14130516324120243245311158864, 7.949432902522207787558221081954, 9.061440982478982806773319922482, 9.704698545769151316221049114939

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