Properties

Label 2-1536-1.1-c1-0-6
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 − 3.95·5-s + 1.63·7-s + 9-s + 4.82·11-s − 5.59·13-s − 3.95·15-s − 0.828·17-s − 2.82·19-s + 1.63·21-s + 7.91·23-s + 10.6·25-s + 27-s + 7.23·29-s − 1.63·31-s + 4.82·33-s − 6.48·35-s + 2.31·37-s − 5.59·39-s + 3.17·41-s + 4.48·43-s − 3.95·45-s + 7.91·47-s − 4.31·49-s − 0.828·51-s + 0.678·53-s − 19.1·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.76·5-s + 0.619·7-s + 0.333·9-s + 1.45·11-s − 1.55·13-s − 1.02·15-s − 0.200·17-s − 0.648·19-s + 0.357·21-s + 1.65·23-s + 2.13·25-s + 0.192·27-s + 1.34·29-s − 0.294·31-s + 0.840·33-s − 1.09·35-s + 0.381·37-s − 0.896·39-s + 0.495·41-s + 0.683·43-s − 0.589·45-s + 1.15·47-s − 0.616·49-s − 0.116·51-s + 0.0932·53-s − 2.57·55-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

\(L(1)\) \(\approx\) \(1.629373107\)
\(L(\frac12)\) \(\approx\) \(1.629373107\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.95T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 - 2.31T + 37T^{2} \)
41 \( 1 - 3.17T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 - 7.91T + 47T^{2} \)
53 \( 1 - 0.678T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 1.63T + 79T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−9.070607675552095609242114780060, −8.754613380528066396220468603874, −7.74833601986276248342257014573, −7.29553132270502361943727247639, −6.54329110643883996842409073755, −4.84740150890825885802562223034, −4.40809179445598129659432166342, −3.53839627562226145909601721397, −2.49457424504648060187593979008, −0.901332249465280244979342675905, 0.901332249465280244979342675905, 2.49457424504648060187593979008, 3.53839627562226145909601721397, 4.40809179445598129659432166342, 4.84740150890825885802562223034, 6.54329110643883996842409073755, 7.29553132270502361943727247639, 7.74833601986276248342257014573, 8.754613380528066396220468603874, 9.070607675552095609242114780060

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