Properties

Label 2-6041-1.1-c1-0-76
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  − 2.71·2-s − 0.898·3-s + 5.39·4-s + 3.58·5-s + 2.44·6-s + 7-s − 9.21·8-s − 2.19·9-s − 9.75·10-s − 6.15·11-s − 4.84·12-s + 0.437·13-s − 2.71·14-s − 3.22·15-s + 14.2·16-s − 4.15·17-s + 5.96·18-s − 2.98·19-s + 19.3·20-s − 0.898·21-s + 16.7·22-s + 3.64·23-s + 8.28·24-s + 7.86·25-s − 1.18·26-s + 4.66·27-s + 5.39·28-s + ⋯
L(s)  = 1  − 1.92·2-s − 0.518·3-s + 2.69·4-s + 1.60·5-s + 0.997·6-s + 0.377·7-s − 3.25·8-s − 0.730·9-s − 3.08·10-s − 1.85·11-s − 1.39·12-s + 0.121·13-s − 0.726·14-s − 0.832·15-s + 3.57·16-s − 1.00·17-s + 1.40·18-s − 0.685·19-s + 4.32·20-s − 0.196·21-s + 3.56·22-s + 0.759·23-s + 1.69·24-s + 1.57·25-s − 0.233·26-s + 0.897·27-s + 1.01·28-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

\(L(1)\) \(\approx\) \(0.5365365666\)
\(L(\frac12)\) \(\approx\) \(0.5365365666\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 + 0.898T + 3T^{2} \)
5 \( 1 - 3.58T + 5T^{2} \)
11 \( 1 + 6.15T + 11T^{2} \)
13 \( 1 - 0.437T + 13T^{2} \)
17 \( 1 + 4.15T + 17T^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 - 3.64T + 23T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 + 0.0845T + 31T^{2} \)
37 \( 1 - 4.52T + 37T^{2} \)
41 \( 1 - 5.05T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 6.30T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 - 0.429T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 4.10T + 79T^{2} \)
83 \( 1 + 1.17T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 2.83T + 97T^{2} \)
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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

−8.173244530233495172373269319969, −7.59031132853040117586472478217, −6.75422233915119218867267408474, −5.99800187417324469275117512294, −5.68735605032395513022170483499, −4.82565893693670964563875983204, −2.88401863853078972429438912118, −2.40017893929178812879971907137, −1.73611917311182210514439664467, −0.50468491968269553223459644335, 0.50468491968269553223459644335, 1.73611917311182210514439664467, 2.40017893929178812879971907137, 2.88401863853078972429438912118, 4.82565893693670964563875983204, 5.68735605032395513022170483499, 5.99800187417324469275117512294, 6.75422233915119218867267408474, 7.59031132853040117586472478217, 8.173244530233495172373269319969

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