Properties

Label 2-1536-1.1-c1-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.08·5-s − 5.03·7-s + 9-s + 0.828·11-s − 2.94·13-s − 2.08·15-s + 4.82·17-s − 2.82·19-s + 5.03·21-s + 4.16·23-s − 0.656·25-s − 27-s + 7.97·29-s + 5.03·31-s − 0.828·33-s − 10.4·35-s − 7.11·37-s + 2.94·39-s + 8.82·41-s + 12.4·43-s + 2.08·45-s + 4.16·47-s + 18.3·49-s − 4.82·51-s − 12.1·53-s + 1.72·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.932·5-s − 1.90·7-s + 0.333·9-s + 0.249·11-s − 0.817·13-s − 0.538·15-s + 1.17·17-s − 0.648·19-s + 1.09·21-s + 0.869·23-s − 0.131·25-s − 0.192·27-s + 1.48·29-s + 0.903·31-s − 0.144·33-s − 1.77·35-s − 1.16·37-s + 0.471·39-s + 1.37·41-s + 1.90·43-s + 0.310·45-s + 0.607·47-s + 2.61·49-s − 0.676·51-s − 1.66·53-s + 0.232·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.247006877\)
\(L(\frac12)\) \(\approx\) \(1.247006877\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 + 5.03T + 7T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 2.94T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 - 5.03T + 31T^{2} \)
37 \( 1 + 7.11T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 + 7.11T + 61T^{2} \)
67 \( 1 - 2.34T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 5.03T + 79T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−9.560608619902973622066465083754, −9.029323475675788720261044883672, −7.66561928811993864006579595255, −6.75450544371424077006569085441, −6.21539427976045565535184383833, −5.58506194691075350789015056474, −4.50442577350625650473955766344, −3.29479208808159758121386928672, −2.44903647131749265476563518079, −0.802431047703179743830455408225, 0.802431047703179743830455408225, 2.44903647131749265476563518079, 3.29479208808159758121386928672, 4.50442577350625650473955766344, 5.58506194691075350789015056474, 6.21539427976045565535184383833, 6.75450544371424077006569085441, 7.66561928811993864006579595255, 9.029323475675788720261044883672, 9.560608619902973622066465083754

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