Properties

Label 2-1536-1.1-c1-0-22
Degree $2$
Conductor $1536$
Sign $-1$
Analytic cond. $12.2650$
Root an. cond. $3.50214$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.41·5-s + 1.41·7-s + 9-s − 2·11-s + 1.41·15-s + 2·17-s − 4·19-s − 1.41·21-s + 2.82·23-s − 2.99·25-s − 27-s + 9.89·29-s − 7.07·31-s + 2·33-s − 2.00·35-s − 8.48·37-s + 6·41-s − 8·43-s − 1.41·45-s + 2.82·47-s − 5·49-s − 2·51-s − 1.41·53-s + 2.82·55-s + 4·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.632·5-s + 0.534·7-s + 0.333·9-s − 0.603·11-s + 0.365·15-s + 0.485·17-s − 0.917·19-s − 0.308·21-s + 0.589·23-s − 0.599·25-s − 0.192·27-s + 1.83·29-s − 1.27·31-s + 0.348·33-s − 0.338·35-s − 1.39·37-s + 0.937·41-s − 1.21·43-s − 0.210·45-s + 0.412·47-s − 0.714·49-s − 0.280·51-s − 0.194·53-s + 0.381·55-s + 0.529·57-s − 1.56·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-1$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 9.89T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.916750607396651948811808976466, −8.199730259054469365479242027210, −7.47711933085206353286468171164, −6.67272090731286239914611731222, −5.67131832040511532506282615930, −4.88681667454852198314461845105, −4.09930114606367679589254130454, −2.95338886371393166452461882867, −1.56463936515313379078387460553, 0, 1.56463936515313379078387460553, 2.95338886371393166452461882867, 4.09930114606367679589254130454, 4.88681667454852198314461845105, 5.67131832040511532506282615930, 6.67272090731286239914611731222, 7.47711933085206353286468171164, 8.199730259054469365479242027210, 8.916750607396651948811808976466

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