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  + 1.17·3-s + 5-s − 3.16·7-s − 1.61·9-s − 0.427·11-s + 0.986·13-s + 1.17·15-s + 2.72·17-s − 1.41·19-s − 3.72·21-s + 6.25·23-s + 25-s − 5.42·27-s + 1.45·29-s + 0.966·31-s − 0.501·33-s − 3.16·35-s − 7.21·37-s + 1.15·39-s − 10.4·41-s + 9.12·43-s − 1.61·45-s − 12.8·47-s + 3.04·49-s + 3.20·51-s − 3.99·53-s − 0.427·55-s + ⋯
L(s)  = 1  + 0.678·3-s + 0.447·5-s − 1.19·7-s − 0.539·9-s − 0.128·11-s + 0.273·13-s + 0.303·15-s + 0.660·17-s − 0.325·19-s − 0.812·21-s + 1.30·23-s + 0.200·25-s − 1.04·27-s + 0.269·29-s + 0.173·31-s − 0.0873·33-s − 0.535·35-s − 1.18·37-s + 0.185·39-s − 1.63·41-s + 1.39·43-s − 0.241·45-s − 1.86·47-s + 0.434·49-s + 0.448·51-s − 0.548·53-s − 0.0575·55-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 \( 1 - 1.17T + 3T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 0.427T + 11T^{2} \)
13 \( 1 - 0.986T + 13T^{2} \)
17 \( 1 - 2.72T + 17T^{2} \)
19 \( 1 + 1.41T + 19T^{2} \)
23 \( 1 - 6.25T + 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 - 0.966T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 3.99T + 53T^{2} \)
59 \( 1 + 0.170T + 59T^{2} \)
61 \( 1 + 0.957T + 61T^{2} \)
67 \( 1 - 1.61T + 67T^{2} \)
71 \( 1 - 7.83T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 6.07T + 83T^{2} \)
89 \( 1 - 5.74T + 89T^{2} \)
97 \( 1 + 5.12T + 97T^{2} \)
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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

−7.85743224951617942636007787097, −6.85921520811239511142014830391, −6.44895946332395579698233493277, −5.58936851645823325043570706946, −4.95072668730483479095928028655, −3.70353065174740640158185634393, −3.17270788425848429169067250007, −2.56706738042353185078031330294, −1.42413722760490484930917916240, 0, 1.42413722760490484930917916240, 2.56706738042353185078031330294, 3.17270788425848429169067250007, 3.70353065174740640158185634393, 4.95072668730483479095928028655, 5.58936851645823325043570706946, 6.44895946332395579698233493277, 6.85921520811239511142014830391, 7.85743224951617942636007787097

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