Properties

Label 2-4056-1.1-c1-0-0
Degree $2$
Conductor $4056$
Sign $1$
Analytic cond. $32.3873$
Root an. cond. $5.69098$
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 − 2.65·5-s − 3.72·7-s + 9-s − 3.03·11-s + 2.65·15-s + 3.27·17-s − 7.35·19-s + 3.72·21-s − 4.11·23-s + 2.06·25-s − 27-s − 0.411·29-s − 5.12·31-s + 3.03·33-s + 9.89·35-s − 9.87·37-s − 1.73·41-s − 10.0·43-s − 2.65·45-s − 1.09·47-s + 6.85·49-s − 3.27·51-s − 13.6·53-s + 8.07·55-s + 7.35·57-s − 12.5·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.18·5-s − 1.40·7-s + 0.333·9-s − 0.915·11-s + 0.686·15-s + 0.795·17-s − 1.68·19-s + 0.812·21-s − 0.857·23-s + 0.413·25-s − 0.192·27-s − 0.0764·29-s − 0.920·31-s + 0.528·33-s + 1.67·35-s − 1.62·37-s − 0.270·41-s − 1.53·43-s − 0.396·45-s − 0.159·47-s + 0.979·49-s − 0.459·51-s − 1.87·53-s + 1.08·55-s + 0.973·57-s − 1.63·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(32.3873\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07353951838\)
\(L(\frac12)\) \(\approx\) \(0.07353951838\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + 2.65T + 5T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
17 \( 1 - 3.27T + 17T^{2} \)
19 \( 1 + 7.35T + 19T^{2} \)
23 \( 1 + 4.11T + 23T^{2} \)
29 \( 1 + 0.411T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 9.87T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 1.09T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 1.36T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 5.45T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 2.40T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.192036530183539678661050887143, −7.80742733968604428853600716901, −6.82414637743092674816831303064, −6.38441449619503469765347185171, −5.48279940550125859955042545402, −4.66381241725809664287736590787, −3.68502392956409687527660663789, −3.27895563752927523578265034261, −1.96274113537274651033660762345, −0.14915558863896510494242838679, 0.14915558863896510494242838679, 1.96274113537274651033660762345, 3.27895563752927523578265034261, 3.68502392956409687527660663789, 4.66381241725809664287736590787, 5.48279940550125859955042545402, 6.38441449619503469765347185171, 6.82414637743092674816831303064, 7.80742733968604428853600716901, 8.192036530183539678661050887143

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