Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 151 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.31·3-s + 5-s − 0.477·7-s + 7.95·9-s + 4.57·11-s + 2.86·13-s − 3.31·15-s + 7.02·17-s − 5.71·19-s + 1.58·21-s − 1.60·23-s + 25-s − 16.4·27-s − 8.07·29-s − 11.0·31-s − 15.1·33-s − 0.477·35-s − 2.92·37-s − 9.49·39-s − 5.80·41-s − 7.00·43-s + 7.95·45-s + 12.4·47-s − 6.77·49-s − 23.2·51-s − 0.250·53-s + 4.57·55-s + ⋯
L(s)  = 1  − 1.91·3-s + 0.447·5-s − 0.180·7-s + 2.65·9-s + 1.37·11-s + 0.795·13-s − 0.854·15-s + 1.70·17-s − 1.31·19-s + 0.344·21-s − 0.334·23-s + 0.200·25-s − 3.15·27-s − 1.49·29-s − 1.97·31-s − 2.63·33-s − 0.0807·35-s − 0.481·37-s − 1.52·39-s − 0.905·41-s − 1.06·43-s + 1.18·45-s + 1.80·47-s − 0.967·49-s − 3.25·51-s − 0.0344·53-s + 0.616·55-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

\( d \)  =  \(2\)
\( N \)  =  \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6040,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;151\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;151\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - T \)
151 \( 1 - T \)
good3 \( 1 + 3.31T + 3T^{2} \)
7 \( 1 + 0.477T + 7T^{2} \)
11 \( 1 - 4.57T + 11T^{2} \)
13 \( 1 - 2.86T + 13T^{2} \)
17 \( 1 - 7.02T + 17T^{2} \)
19 \( 1 + 5.71T + 19T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + 8.07T + 29T^{2} \)
31 \( 1 + 11.0T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 0.250T + 53T^{2} \)
59 \( 1 - 2.86T + 59T^{2} \)
61 \( 1 - 2.93T + 61T^{2} \)
67 \( 1 - 0.359T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 6.69T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 4.08T + 83T^{2} \)
89 \( 1 + 2.51T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.37723336762683207814474082624, −6.76437954445814204070107002195, −6.22999373044116645796571433479, −5.58777664892392364916091588867, −5.21076739593661041090491637929, −3.92248935156280505809378007264, −3.75209336387424291607077232325, −1.80201326032035768013702380445, −1.25616340787946005613562024557, 0, 1.25616340787946005613562024557, 1.80201326032035768013702380445, 3.75209336387424291607077232325, 3.92248935156280505809378007264, 5.21076739593661041090491637929, 5.58777664892392364916091588867, 6.22999373044116645796571433479, 6.76437954445814204070107002195, 7.37723336762683207814474082624

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