Properties

Label 2-6039-1.1-c1-0-125
Degree $2$
Conductor $6039$
Sign $-1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0892·2-s − 1.99·4-s − 0.250·5-s − 4.16·7-s + 0.356·8-s + 0.0223·10-s − 11-s − 2.99·13-s + 0.371·14-s + 3.95·16-s + 4.74·17-s − 0.0676·19-s + 0.499·20-s + 0.0892·22-s + 5.30·23-s − 4.93·25-s + 0.267·26-s + 8.29·28-s + 5.69·29-s − 1.66·31-s − 1.06·32-s − 0.423·34-s + 1.04·35-s − 3.66·37-s + 0.00604·38-s − 0.0893·40-s − 1.93·41-s + ⋯
L(s)  = 1  − 0.0631·2-s − 0.996·4-s − 0.112·5-s − 1.57·7-s + 0.125·8-s + 0.00707·10-s − 0.301·11-s − 0.830·13-s + 0.0993·14-s + 0.988·16-s + 1.15·17-s − 0.0155·19-s + 0.111·20-s + 0.0190·22-s + 1.10·23-s − 0.987·25-s + 0.0524·26-s + 1.56·28-s + 1.05·29-s − 0.299·31-s − 0.188·32-s − 0.0726·34-s + 0.176·35-s − 0.602·37-s + 0.000980·38-s − 0.0141·40-s − 0.301·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
61 \( 1 + T \)
good2 \( 1 + 0.0892T + 2T^{2} \)
5 \( 1 + 0.250T + 5T^{2} \)
7 \( 1 + 4.16T + 7T^{2} \)
13 \( 1 + 2.99T + 13T^{2} \)
17 \( 1 - 4.74T + 17T^{2} \)
19 \( 1 + 0.0676T + 19T^{2} \)
23 \( 1 - 5.30T + 23T^{2} \)
29 \( 1 - 5.69T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 + 3.66T + 37T^{2} \)
41 \( 1 + 1.93T + 41T^{2} \)
43 \( 1 - 8.77T + 43T^{2} \)
47 \( 1 - 7.56T + 47T^{2} \)
53 \( 1 - 6.58T + 53T^{2} \)
59 \( 1 + 6.14T + 59T^{2} \)
67 \( 1 - 3.02T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 7.54T + 83T^{2} \)
89 \( 1 + 3.77T + 89T^{2} \)
97 \( 1 - 1.83T + 97T^{2} \)
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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

−7.60942846982481472594882060668, −7.19695270406625163168786933974, −6.19292027968582641367228175613, −5.56980381496979849125166902107, −4.85942968850467145654469572598, −3.95151929010726291597750674281, −3.29317862073184510406310496499, −2.57226579784343075901360257662, −0.975377770634200202218769318294, 0, 0.975377770634200202218769318294, 2.57226579784343075901360257662, 3.29317862073184510406310496499, 3.95151929010726291597750674281, 4.85942968850467145654469572598, 5.56980381496979849125166902107, 6.19292027968582641367228175613, 7.19695270406625163168786933974, 7.60942846982481472594882060668

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