Properties

Label 2-6039-1.1-c1-0-163
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  − 1.32·2-s − 0.255·4-s − 0.125·5-s + 1.14·7-s + 2.97·8-s + 0.165·10-s + 11-s − 2.50·13-s − 1.51·14-s − 3.42·16-s − 3.85·17-s + 6.27·19-s + 0.0319·20-s − 1.32·22-s + 3.82·23-s − 4.98·25-s + 3.30·26-s − 0.292·28-s + 1.92·29-s − 3.36·31-s − 1.43·32-s + 5.09·34-s − 0.143·35-s − 4.87·37-s − 8.29·38-s − 0.372·40-s + 12.3·41-s + ⋯
L(s)  = 1  − 0.933·2-s − 0.127·4-s − 0.0559·5-s + 0.433·7-s + 1.05·8-s + 0.0522·10-s + 0.301·11-s − 0.694·13-s − 0.404·14-s − 0.855·16-s − 0.934·17-s + 1.44·19-s + 0.00714·20-s − 0.281·22-s + 0.797·23-s − 0.996·25-s + 0.648·26-s − 0.0553·28-s + 0.356·29-s − 0.603·31-s − 0.253·32-s + 0.872·34-s − 0.0242·35-s − 0.800·37-s − 1.34·38-s − 0.0589·40-s + 1.92·41-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 \( 1 + 1.32T + 2T^{2} \)
5 \( 1 + 0.125T + 5T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
13 \( 1 + 2.50T + 13T^{2} \)
17 \( 1 + 3.85T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 - 3.82T + 23T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 + 3.36T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 5.75T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 - 6.57T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
73 \( 1 + 8.55T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 9.44T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−7.61242960002150766568093683903, −7.42763704775216258763768174990, −6.52235441595635742878156722305, −5.47156766976465547623581344162, −4.82431053511558601316681579286, −4.16120434664477626443187836175, −3.13438324830162280986225731917, −2.02537223009966857607732510646, −1.17534428993669592335549457052, 0, 1.17534428993669592335549457052, 2.02537223009966857607732510646, 3.13438324830162280986225731917, 4.16120434664477626443187836175, 4.82431053511558601316681579286, 5.47156766976465547623581344162, 6.52235441595635742878156722305, 7.42763704775216258763768174990, 7.61242960002150766568093683903

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