Properties

Label 2-1536-1.1-c1-0-11
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 + 3.41·5-s − 1.41·7-s + 9-s − 4.82·11-s − 0.828·13-s + 3.41·15-s + 4.82·17-s + 2.82·19-s − 1.41·21-s + 1.17·23-s + 6.65·25-s + 27-s + 7.41·29-s + 7.07·31-s − 4.82·33-s − 4.82·35-s + 11.6·37-s − 0.828·39-s − 10.4·41-s + 6.82·43-s + 3.41·45-s + 12.4·47-s − 5·49-s + 4.82·51-s + 1.75·53-s − 16.4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.52·5-s − 0.534·7-s + 0.333·9-s − 1.45·11-s − 0.229·13-s + 0.881·15-s + 1.17·17-s + 0.648·19-s − 0.308·21-s + 0.244·23-s + 1.33·25-s + 0.192·27-s + 1.37·29-s + 1.27·31-s − 0.840·33-s − 0.816·35-s + 1.91·37-s − 0.132·39-s − 1.63·41-s + 1.04·43-s + 0.508·45-s + 1.82·47-s − 0.714·49-s + 0.676·51-s + 0.241·53-s − 2.22·55-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

\(L(1)\) \(\approx\) \(2.591232465\)
\(L(\frac12)\) \(\approx\) \(2.591232465\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 - 7.41T + 29T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 6.82T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 1.75T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 - 5.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−9.694468455169911339648426407487, −8.726242909827584481788428541958, −7.902143102458363450844213594938, −7.08914896736604951814659432982, −6.03927125108928665263293464136, −5.48865383006853673783369906253, −4.52854011525196972213481184143, −2.88428709462737857148108864879, −2.67676023609515521262976787307, −1.20272965852503917689697692874, 1.20272965852503917689697692874, 2.67676023609515521262976787307, 2.88428709462737857148108864879, 4.52854011525196972213481184143, 5.48865383006853673783369906253, 6.03927125108928665263293464136, 7.08914896736604951814659432982, 7.902143102458363450844213594938, 8.726242909827584481788428541958, 9.694468455169911339648426407487

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