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  + 2.50·3-s + 5-s − 2.23·7-s + 3.25·9-s − 0.595·11-s − 3.72·13-s + 2.50·15-s − 2.87·17-s − 4.54·19-s − 5.58·21-s + 6.52·23-s + 25-s + 0.634·27-s − 3.40·29-s − 4.46·31-s − 1.48·33-s − 2.23·35-s − 1.19·37-s − 9.31·39-s + 6.99·41-s − 6.50·43-s + 3.25·45-s + 1.73·47-s − 2.00·49-s − 7.19·51-s + 1.40·53-s − 0.595·55-s + ⋯
L(s)  = 1  + 1.44·3-s + 0.447·5-s − 0.844·7-s + 1.08·9-s − 0.179·11-s − 1.03·13-s + 0.645·15-s − 0.697·17-s − 1.04·19-s − 1.21·21-s + 1.35·23-s + 0.200·25-s + 0.122·27-s − 0.632·29-s − 0.801·31-s − 0.259·33-s − 0.377·35-s − 0.197·37-s − 1.49·39-s + 1.09·41-s − 0.991·43-s + 0.485·45-s + 0.252·47-s − 0.286·49-s − 1.00·51-s + 0.192·53-s − 0.0803·55-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 \( 1 - 2.50T + 3T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
11 \( 1 + 0.595T + 11T^{2} \)
13 \( 1 + 3.72T + 13T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 + 4.54T + 19T^{2} \)
23 \( 1 - 6.52T + 23T^{2} \)
29 \( 1 + 3.40T + 29T^{2} \)
31 \( 1 + 4.46T + 31T^{2} \)
37 \( 1 + 1.19T + 37T^{2} \)
41 \( 1 - 6.99T + 41T^{2} \)
43 \( 1 + 6.50T + 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 - 1.40T + 53T^{2} \)
59 \( 1 - 4.59T + 59T^{2} \)
61 \( 1 + 0.884T + 61T^{2} \)
67 \( 1 + 7.08T + 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 5.17T + 79T^{2} \)
83 \( 1 + 1.25T + 83T^{2} \)
89 \( 1 + 0.496T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−7.67599810439103221898042932743, −7.15714776073075401770738406504, −6.49404386003611836324481817773, −5.58422677153683748734210399718, −4.67365912486107403837117886806, −3.87990357094745073975048006281, −2.99331352791671227635514808124, −2.52081227440498712392612148022, −1.70102033360078356689653026280, 0, 1.70102033360078356689653026280, 2.52081227440498712392612148022, 2.99331352791671227635514808124, 3.87990357094745073975048006281, 4.67365912486107403837117886806, 5.58422677153683748734210399718, 6.49404386003611836324481817773, 7.15714776073075401770738406504, 7.67599810439103221898042932743

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