Properties

Label 2-6041-1.1-c1-0-138
Degree $2$
Conductor $6041$
Sign $1$
Analytic cond. $48.2376$
Root an. cond. $6.94533$
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  − 2.62·2-s − 1.27·3-s + 4.88·4-s + 1.41·5-s + 3.33·6-s + 7-s − 7.56·8-s − 1.38·9-s − 3.72·10-s + 1.00·11-s − 6.20·12-s + 2.84·13-s − 2.62·14-s − 1.80·15-s + 10.0·16-s − 0.358·17-s + 3.62·18-s + 5.52·19-s + 6.92·20-s − 1.27·21-s − 2.63·22-s − 1.69·23-s + 9.61·24-s − 2.98·25-s − 7.45·26-s + 5.57·27-s + 4.88·28-s + ⋯
L(s)  = 1  − 1.85·2-s − 0.734·3-s + 2.44·4-s + 0.634·5-s + 1.36·6-s + 0.377·7-s − 2.67·8-s − 0.461·9-s − 1.17·10-s + 0.302·11-s − 1.79·12-s + 0.787·13-s − 0.701·14-s − 0.465·15-s + 2.52·16-s − 0.0869·17-s + 0.855·18-s + 1.26·19-s + 1.54·20-s − 0.277·21-s − 0.561·22-s − 0.354·23-s + 1.96·24-s − 0.597·25-s − 1.46·26-s + 1.07·27-s + 0.922·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $1$
Analytic conductor: \(48.2376\)
Root analytic conductor: \(6.94533\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6041,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7961432041\)
\(L(\frac12)\) \(\approx\) \(0.7961432041\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
863 \( 1 + T \)
good2 \( 1 + 2.62T + 2T^{2} \)
3 \( 1 + 1.27T + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 - 2.84T + 13T^{2} \)
17 \( 1 + 0.358T + 17T^{2} \)
19 \( 1 - 5.52T + 19T^{2} \)
23 \( 1 + 1.69T + 23T^{2} \)
29 \( 1 - 5.52T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 - 7.60T + 37T^{2} \)
41 \( 1 - 7.89T + 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 0.388T + 59T^{2} \)
61 \( 1 - 4.85T + 61T^{2} \)
67 \( 1 + 8.67T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 2.15T + 79T^{2} \)
83 \( 1 - 8.98T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−8.025412655777920745716837099089, −7.73328716148518438160279030140, −6.58682657625689480394642170505, −6.23468689835074086452548199202, −5.63807003156939551103433552369, −4.62469992439406741562036303397, −3.22655124562232747460184566705, −2.39672023488953165143105256122, −1.38093201096877405110550289899, −0.71524365555508019206886531501, 0.71524365555508019206886531501, 1.38093201096877405110550289899, 2.39672023488953165143105256122, 3.22655124562232747460184566705, 4.62469992439406741562036303397, 5.63807003156939551103433552369, 6.23468689835074086452548199202, 6.58682657625689480394642170505, 7.73328716148518438160279030140, 8.025412655777920745716837099089

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