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  + 0.942·3-s + 5-s + 1.85·7-s − 2.11·9-s − 5.46·11-s − 1.73·13-s + 0.942·15-s + 7.05·17-s − 0.773·19-s + 1.75·21-s − 3.81·23-s + 25-s − 4.81·27-s − 5.07·29-s − 3.44·31-s − 5.14·33-s + 1.85·35-s + 10.1·37-s − 1.62·39-s + 10.9·41-s − 11.4·43-s − 2.11·45-s + 4.80·47-s − 3.54·49-s + 6.64·51-s − 11.9·53-s − 5.46·55-s + ⋯
L(s)  = 1  + 0.543·3-s + 0.447·5-s + 0.702·7-s − 0.704·9-s − 1.64·11-s − 0.479·13-s + 0.243·15-s + 1.71·17-s − 0.177·19-s + 0.382·21-s − 0.794·23-s + 0.200·25-s − 0.926·27-s − 0.943·29-s − 0.618·31-s − 0.896·33-s + 0.314·35-s + 1.66·37-s − 0.260·39-s + 1.71·41-s − 1.73·43-s − 0.314·45-s + 0.701·47-s − 0.505·49-s + 0.930·51-s − 1.63·53-s − 0.736·55-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 \( 1 - 0.942T + 3T^{2} \)
7 \( 1 - 1.85T + 7T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 - 7.05T + 17T^{2} \)
19 \( 1 + 0.773T + 19T^{2} \)
23 \( 1 + 3.81T + 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + 3.44T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 9.86T + 59T^{2} \)
61 \( 1 + 2.97T + 61T^{2} \)
67 \( 1 - 1.49T + 67T^{2} \)
71 \( 1 + 4.03T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 6.78T + 79T^{2} \)
83 \( 1 + 8.89T + 83T^{2} \)
89 \( 1 - 9.29T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−7.939201015893486506339635928409, −7.35895995573256889954960801222, −6.04729385139709091640029046476, −5.54381012337474825777921385084, −5.02377005218618240146915506411, −3.98747139043815404272438732530, −2.96262838258283539336315037369, −2.48137323739428011561686108998, −1.51694611069969245260855497706, 0, 1.51694611069969245260855497706, 2.48137323739428011561686108998, 2.96262838258283539336315037369, 3.98747139043815404272438732530, 5.02377005218618240146915506411, 5.54381012337474825777921385084, 6.04729385139709091640029046476, 7.35895995573256889954960801222, 7.939201015893486506339635928409

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