Properties

Label 2-6042-1.1-c1-0-32
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 − 2.45·5-s + 6-s − 4.70·7-s − 8-s + 9-s + 2.45·10-s + 1.91·11-s − 12-s − 5.90·13-s + 4.70·14-s + 2.45·15-s + 16-s − 1.14·17-s − 18-s − 19-s − 2.45·20-s + 4.70·21-s − 1.91·22-s + 3.30·23-s + 24-s + 1.04·25-s + 5.90·26-s − 27-s − 4.70·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.09·5-s + 0.408·6-s − 1.77·7-s − 0.353·8-s + 0.333·9-s + 0.777·10-s + 0.575·11-s − 0.288·12-s − 1.63·13-s + 1.25·14-s + 0.635·15-s + 0.250·16-s − 0.277·17-s − 0.235·18-s − 0.229·19-s − 0.549·20-s + 1.02·21-s − 0.407·22-s + 0.689·23-s + 0.204·24-s + 0.209·25-s + 1.15·26-s − 0.192·27-s − 0.888·28-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 \( 1 + 2.45T + 5T^{2} \)
7 \( 1 + 4.70T + 7T^{2} \)
11 \( 1 - 1.91T + 11T^{2} \)
13 \( 1 + 5.90T + 13T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
23 \( 1 - 3.30T + 23T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 - 7.67T + 31T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 - 6.19T + 41T^{2} \)
43 \( 1 + 5.22T + 43T^{2} \)
47 \( 1 - 5.78T + 47T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 3.23T + 67T^{2} \)
71 \( 1 - 1.22T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 5.90T + 89T^{2} \)
97 \( 1 + 0.0999T + 97T^{2} \)
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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

−7.61262924924444485304233765218, −6.85598902589973812856495395179, −6.72232141254836034303145325060, −5.77588449713177141354507013608, −4.76709574892509540961070757822, −3.99009705631227732170649182659, −3.15744680209677214399698897151, −2.39801207024205425817159112332, −0.77803566384066747937630543196, 0, 0.77803566384066747937630543196, 2.39801207024205425817159112332, 3.15744680209677214399698897151, 3.99009705631227732170649182659, 4.76709574892509540961070757822, 5.77588449713177141354507013608, 6.72232141254836034303145325060, 6.85598902589973812856495395179, 7.61262924924444485304233765218

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