Properties

Label 2-6034-1.1-c1-0-156
Degree $2$
Conductor $6034$
Sign $-1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 − 2.73·3-s + 4-s + 3.55·5-s + 2.73·6-s + 7-s − 8-s + 4.47·9-s − 3.55·10-s + 1.87·11-s − 2.73·12-s − 2.49·13-s − 14-s − 9.72·15-s + 16-s + 3.64·17-s − 4.47·18-s − 2.29·19-s + 3.55·20-s − 2.73·21-s − 1.87·22-s + 3.29·23-s + 2.73·24-s + 7.65·25-s + 2.49·26-s − 4.04·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.57·3-s + 0.5·4-s + 1.59·5-s + 1.11·6-s + 0.377·7-s − 0.353·8-s + 1.49·9-s − 1.12·10-s + 0.566·11-s − 0.789·12-s − 0.691·13-s − 0.267·14-s − 2.51·15-s + 0.250·16-s + 0.883·17-s − 1.05·18-s − 0.525·19-s + 0.795·20-s − 0.596·21-s − 0.400·22-s + 0.686·23-s + 0.558·24-s + 1.53·25-s + 0.489·26-s − 0.778·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $-1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
431 \( 1 - T \)
good3 \( 1 + 2.73T + 3T^{2} \)
5 \( 1 - 3.55T + 5T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 2.49T + 13T^{2} \)
17 \( 1 - 3.64T + 17T^{2} \)
19 \( 1 + 2.29T + 19T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + 9.17T + 37T^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 - 3.93T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 - 2.09T + 59T^{2} \)
61 \( 1 + 6.35T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.35069009299673128446141535001, −7.03575320459284791091186810186, −6.19076873073951768134345499869, −5.61407007721134344452155668821, −5.28012929712392818796453596708, −4.33556328595756940268440083445, −2.98333678623465459640297916746, −1.78112774623408782674305205532, −1.34819867416438774154270393488, 0, 1.34819867416438774154270393488, 1.78112774623408782674305205532, 2.98333678623465459640297916746, 4.33556328595756940268440083445, 5.28012929712392818796453596708, 5.61407007721134344452155668821, 6.19076873073951768134345499869, 7.03575320459284791091186810186, 7.35069009299673128446141535001

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