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  + 1.76·3-s + 5-s + 0.0387·7-s + 0.122·9-s − 4.56·11-s + 2.95·13-s + 1.76·15-s − 3.20·17-s + 2.27·19-s + 0.0685·21-s + 5.32·23-s + 25-s − 5.08·27-s − 9.46·29-s − 8.25·31-s − 8.07·33-s + 0.0387·35-s − 8.02·37-s + 5.22·39-s − 1.45·41-s − 4.90·43-s + 0.122·45-s − 6.28·47-s − 6.99·49-s − 5.65·51-s − 0.800·53-s − 4.56·55-s + ⋯
L(s)  = 1  + 1.02·3-s + 0.447·5-s + 0.0146·7-s + 0.0408·9-s − 1.37·11-s + 0.820·13-s + 0.456·15-s − 0.776·17-s + 0.522·19-s + 0.0149·21-s + 1.11·23-s + 0.200·25-s − 0.978·27-s − 1.75·29-s − 1.48·31-s − 1.40·33-s + 0.00655·35-s − 1.31·37-s + 0.837·39-s − 0.227·41-s − 0.747·43-s + 0.0182·45-s − 0.916·47-s − 0.999·49-s − 0.792·51-s − 0.110·53-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 \( 1 - 1.76T + 3T^{2} \)
7 \( 1 - 0.0387T + 7T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 + 8.25T + 31T^{2} \)
37 \( 1 + 8.02T + 37T^{2} \)
41 \( 1 + 1.45T + 41T^{2} \)
43 \( 1 + 4.90T + 43T^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 + 0.800T + 53T^{2} \)
59 \( 1 + 7.10T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 2.60T + 83T^{2} \)
89 \( 1 + 1.97T + 89T^{2} \)
97 \( 1 - 6.49T + 97T^{2} \)
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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

−7.83811297241010953493624726345, −7.17551668568137699970516772941, −6.36128946714073868070235652953, −5.34757963044655640905909114863, −5.08447249167320734230836130072, −3.66732300025183849265516371532, −3.27977980662433520073109608007, −2.31668929075883667215564078898, −1.67294659163830924722304616437, 0, 1.67294659163830924722304616437, 2.31668929075883667215564078898, 3.27977980662433520073109608007, 3.66732300025183849265516371532, 5.08447249167320734230836130072, 5.34757963044655640905909114863, 6.36128946714073868070235652953, 7.17551668568137699970516772941, 7.83811297241010953493624726345

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