Properties

Label 2-6042-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 1.35·5-s − 6-s − 2.98·7-s + 8-s + 9-s − 1.35·10-s + 0.528·11-s − 12-s − 2.16·13-s − 2.98·14-s + 1.35·15-s + 16-s − 5.01·17-s + 18-s − 19-s − 1.35·20-s + 2.98·21-s + 0.528·22-s − 0.879·23-s − 24-s − 3.17·25-s − 2.16·26-s − 27-s − 2.98·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.604·5-s − 0.408·6-s − 1.12·7-s + 0.353·8-s + 0.333·9-s − 0.427·10-s + 0.159·11-s − 0.288·12-s − 0.600·13-s − 0.798·14-s + 0.348·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.302·20-s + 0.652·21-s + 0.112·22-s − 0.183·23-s − 0.204·24-s − 0.634·25-s − 0.424·26-s − 0.192·27-s − 0.564·28-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: \(0\)
Selberg data: \((2,\ 6042,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218515451\)
\(L(\frac12)\) \(\approx\) \(1.218515451\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 - 0.528T + 11T^{2} \)
13 \( 1 + 2.16T + 13T^{2} \)
17 \( 1 + 5.01T + 17T^{2} \)
23 \( 1 + 0.879T + 23T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 - 2.45T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 - 5.66T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 9.72T + 47T^{2} \)
59 \( 1 + 3.89T + 59T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 - 6.77T + 73T^{2} \)
79 \( 1 + 7.58T + 79T^{2} \)
83 \( 1 - 7.33T + 83T^{2} \)
89 \( 1 - 7.98T + 89T^{2} \)
97 \( 1 + 4.09T + 97T^{2} \)
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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

−7.70461005783251736296334126887, −7.31052895046248891575721249896, −6.37350022948226356735632673526, −6.13585683058407470401570809772, −5.16946959959803650441086630915, −4.30482564358872240856618122010, −3.89412378143426290757339422849, −2.89736289068448868344765257574, −2.06641500794090967707216668728, −0.50719763214068821597426391475, 0.50719763214068821597426391475, 2.06641500794090967707216668728, 2.89736289068448868344765257574, 3.89412378143426290757339422849, 4.30482564358872240856618122010, 5.16946959959803650441086630915, 6.13585683058407470401570809772, 6.37350022948226356735632673526, 7.31052895046248891575721249896, 7.70461005783251736296334126887

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