Properties

Label 2-2e12-1.1-c1-0-39
Degree $2$
Conductor $4096$
Sign $1$
Analytic cond. $32.7067$
Root an. cond. $5.71897$
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  − 0.765·3-s + 2.93·5-s − 1.41·7-s − 2.41·9-s + 4.46·11-s + 0.765·13-s − 2.24·15-s − 2.82·17-s + 4.01·19-s + 1.08·21-s + 8.24·23-s + 3.58·25-s + 4.14·27-s − 3.37·29-s − 4·31-s − 3.41·33-s − 4.14·35-s − 0.765·37-s − 0.585·39-s + 0.242·41-s + 5.09·43-s − 7.07·45-s + 0.343·47-s − 5·49-s + 2.16·51-s − 1.21·53-s + 13.0·55-s + ⋯
L(s)  = 1  − 0.441·3-s + 1.31·5-s − 0.534·7-s − 0.804·9-s + 1.34·11-s + 0.212·13-s − 0.579·15-s − 0.685·17-s + 0.920·19-s + 0.236·21-s + 1.71·23-s + 0.717·25-s + 0.797·27-s − 0.627·29-s − 0.718·31-s − 0.594·33-s − 0.700·35-s − 0.125·37-s − 0.0938·39-s + 0.0378·41-s + 0.776·43-s − 1.05·45-s + 0.0500·47-s − 0.714·49-s + 0.303·51-s − 0.166·53-s + 1.76·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(32.7067\)
Root analytic conductor: \(5.71897\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4096,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.059975684\)
\(L(\frac12)\) \(\approx\) \(2.059975684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 0.765T + 3T^{2} \)
5 \( 1 - 2.93T + 5T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 - 4.46T + 11T^{2} \)
13 \( 1 - 0.765T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 4.01T + 19T^{2} \)
23 \( 1 - 8.24T + 23T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 0.765T + 37T^{2} \)
41 \( 1 - 0.242T + 41T^{2} \)
43 \( 1 - 5.09T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 1.21T + 53T^{2} \)
59 \( 1 - 4.90T + 59T^{2} \)
61 \( 1 + 1.84T + 61T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 - 8.24T + 71T^{2} \)
73 \( 1 - 9.89T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 4.90T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−8.856313051979487730953540238596, −7.53414710337617039749418878705, −6.63179091674743259614464418007, −6.30583384465341812978344227872, −5.52484016574616715552493761348, −4.94928562724744873809332586146, −3.72550579344052382439544066774, −2.92815817530618094709578774489, −1.90088098073121330925499240519, −0.861250673588195133176180452833, 0.861250673588195133176180452833, 1.90088098073121330925499240519, 2.92815817530618094709578774489, 3.72550579344052382439544066774, 4.94928562724744873809332586146, 5.52484016574616715552493761348, 6.30583384465341812978344227872, 6.63179091674743259614464418007, 7.53414710337617039749418878705, 8.856313051979487730953540238596

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