Properties

Label 2-2e12-1.1-c1-0-20
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  + 0.482·3-s − 1.47·5-s + 0.191·7-s − 2.76·9-s − 3.38·11-s − 2.48·13-s − 0.713·15-s + 3.11·17-s + 6.49·19-s + 0.0921·21-s − 7.33·23-s − 2.81·25-s − 2.78·27-s + 4.69·29-s − 7.44·31-s − 1.63·33-s − 0.282·35-s + 9.13·37-s − 1.19·39-s + 6.04·41-s + 4.68·43-s + 4.09·45-s + 12.0·47-s − 6.96·49-s + 1.50·51-s − 3.70·53-s + 4.99·55-s + ⋯
L(s)  = 1  + 0.278·3-s − 0.661·5-s + 0.0722·7-s − 0.922·9-s − 1.01·11-s − 0.689·13-s − 0.184·15-s + 0.754·17-s + 1.48·19-s + 0.0201·21-s − 1.52·23-s − 0.562·25-s − 0.535·27-s + 0.870·29-s − 1.33·31-s − 0.283·33-s − 0.0477·35-s + 1.50·37-s − 0.192·39-s + 0.944·41-s + 0.715·43-s + 0.609·45-s + 1.76·47-s − 0.994·49-s + 0.210·51-s − 0.509·53-s + 0.674·55-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

\(L(1)\) \(\approx\) \(1.219233817\)
\(L(\frac12)\) \(\approx\) \(1.219233817\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.482T + 3T^{2} \)
5 \( 1 + 1.47T + 5T^{2} \)
7 \( 1 - 0.191T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + 2.48T + 13T^{2} \)
17 \( 1 - 3.11T + 17T^{2} \)
19 \( 1 - 6.49T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 - 9.13T + 37T^{2} \)
41 \( 1 - 6.04T + 41T^{2} \)
43 \( 1 - 4.68T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 3.70T + 53T^{2} \)
59 \( 1 - 3.04T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 + 4.04T + 71T^{2} \)
73 \( 1 + 3.53T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 7.02T + 89T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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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

−8.053743541074625224832446510406, −7.85408883729462157032525347256, −7.33547120348551944034297027015, −5.97753341604668377929996315816, −5.56440583256556361601164195560, −4.66284740379857498037578675911, −3.71896683864924389755090638098, −2.95201165007705980914780842954, −2.18728190254734097045691701493, −0.59281701989059651385653285096, 0.59281701989059651385653285096, 2.18728190254734097045691701493, 2.95201165007705980914780842954, 3.71896683864924389755090638098, 4.66284740379857498037578675911, 5.56440583256556361601164195560, 5.97753341604668377929996315816, 7.33547120348551944034297027015, 7.85408883729462157032525347256, 8.053743541074625224832446510406

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