Properties

Label 2-6040-1.1-c1-0-1
Degree $2$
Conductor $6040$
Sign $1$
Analytic cond. $48.2296$
Root an. cond. $6.94475$
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  − 1.89·3-s − 5-s − 1.82·7-s + 0.594·9-s − 1.31·11-s − 1.28·13-s + 1.89·15-s − 4.14·17-s − 1.85·19-s + 3.45·21-s + 1.25·23-s + 25-s + 4.56·27-s − 0.373·29-s − 4.20·31-s + 2.49·33-s + 1.82·35-s − 2.64·37-s + 2.44·39-s − 10.4·41-s − 0.450·43-s − 0.594·45-s − 4.47·47-s − 3.68·49-s + 7.85·51-s − 7.87·53-s + 1.31·55-s + ⋯
L(s)  = 1  − 1.09·3-s − 0.447·5-s − 0.687·7-s + 0.198·9-s − 0.397·11-s − 0.357·13-s + 0.489·15-s − 1.00·17-s − 0.426·19-s + 0.753·21-s + 0.262·23-s + 0.200·25-s + 0.877·27-s − 0.0694·29-s − 0.754·31-s + 0.435·33-s + 0.307·35-s − 0.435·37-s + 0.391·39-s − 1.63·41-s − 0.0686·43-s − 0.0886·45-s − 0.652·47-s − 0.526·49-s + 1.10·51-s − 1.08·53-s + 0.177·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
Sign: $1$
Analytic conductor: \(48.2296\)
Root analytic conductor: \(6.94475\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1819558138\)
\(L(\frac12)\) \(\approx\) \(0.1819558138\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
151 \( 1 - T \)
good3 \( 1 + 1.89T + 3T^{2} \)
7 \( 1 + 1.82T + 7T^{2} \)
11 \( 1 + 1.31T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + 0.373T + 29T^{2} \)
31 \( 1 + 4.20T + 31T^{2} \)
37 \( 1 + 2.64T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.450T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 + 7.87T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 - 6.34T + 71T^{2} \)
73 \( 1 - 0.378T + 73T^{2} \)
79 \( 1 - 6.71T + 79T^{2} \)
83 \( 1 - 5.89T + 83T^{2} \)
89 \( 1 - 5.92T + 89T^{2} \)
97 \( 1 + 7.69T + 97T^{2} \)
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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

−8.001232123316057664733484546112, −7.21661735630387470275320893817, −6.46624216950511678920592680160, −6.15408596500940495966369625398, −5.01922758394693935276858915424, −4.78765830861961135736511377794, −3.63571289445564909455843447863, −2.89429627741745407965897860216, −1.73704891052502718138443343572, −0.22748610789417452379423566081, 0.22748610789417452379423566081, 1.73704891052502718138443343572, 2.89429627741745407965897860216, 3.63571289445564909455843447863, 4.78765830861961135736511377794, 5.01922758394693935276858915424, 6.15408596500940495966369625398, 6.46624216950511678920592680160, 7.21661735630387470275320893817, 8.001232123316057664733484546112

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