Properties

Label 2-6042-1.1-c1-0-97
Degree $2$
Conductor $6042$
Sign $-1$
Analytic cond. $48.2456$
Root an. cond. $6.94590$
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  − 2-s − 3-s + 4-s + 2.89·5-s + 6-s − 4.43·7-s − 8-s + 9-s − 2.89·10-s + 2.55·11-s − 12-s + 1.38·13-s + 4.43·14-s − 2.89·15-s + 16-s − 0.833·17-s − 18-s − 19-s + 2.89·20-s + 4.43·21-s − 2.55·22-s + 1.26·23-s + 24-s + 3.36·25-s − 1.38·26-s − 27-s − 4.43·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.29·5-s + 0.408·6-s − 1.67·7-s − 0.353·8-s + 0.333·9-s − 0.914·10-s + 0.769·11-s − 0.288·12-s + 0.384·13-s + 1.18·14-s − 0.746·15-s + 0.250·16-s − 0.202·17-s − 0.235·18-s − 0.229·19-s + 0.646·20-s + 0.967·21-s − 0.543·22-s + 0.262·23-s + 0.204·24-s + 0.673·25-s − 0.272·26-s − 0.192·27-s − 0.837·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6042\)    =    \(2 \cdot 3 \cdot 19 \cdot 53\)
Sign: $-1$
Analytic conductor: \(48.2456\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
19 \( 1 + T \)
53 \( 1 + T \)
good5 \( 1 - 2.89T + 5T^{2} \)
7 \( 1 + 4.43T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 - 1.38T + 13T^{2} \)
17 \( 1 + 0.833T + 17T^{2} \)
23 \( 1 - 1.26T + 23T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + 5.61T + 31T^{2} \)
37 \( 1 + 6.56T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 4.05T + 43T^{2} \)
47 \( 1 + 9.52T + 47T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 7.92T + 67T^{2} \)
71 \( 1 - 4.38T + 71T^{2} \)
73 \( 1 - 7.83T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 4.76T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + 5.66T + 97T^{2} \)
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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

−7.57081783880349660786468964502, −6.77933955200690304092113055584, −6.30632748455264925331493706254, −5.97468305823417838224387126390, −5.09142745008884671730255313072, −3.88251342939149774303020390791, −3.10589458083863582524764841741, −2.12292192997046325534049130455, −1.22232141594391692347187308266, 0, 1.22232141594391692347187308266, 2.12292192997046325534049130455, 3.10589458083863582524764841741, 3.88251342939149774303020390791, 5.09142745008884671730255313072, 5.97468305823417838224387126390, 6.30632748455264925331493706254, 6.77933955200690304092113055584, 7.57081783880349660786468964502

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