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  − 1.96·3-s + 5-s − 2.16·7-s + 0.851·9-s + 3.01·11-s − 2.71·13-s − 1.96·15-s + 0.748·17-s − 4.29·19-s + 4.25·21-s + 1.71·23-s + 25-s + 4.21·27-s − 1.74·29-s + 4.16·31-s − 5.91·33-s − 2.16·35-s + 4.69·37-s + 5.32·39-s − 9.89·41-s + 8.49·43-s + 0.851·45-s + 12.8·47-s − 2.29·49-s − 1.46·51-s − 7.88·53-s + 3.01·55-s + ⋯
L(s)  = 1  − 1.13·3-s + 0.447·5-s − 0.819·7-s + 0.283·9-s + 0.908·11-s − 0.752·13-s − 0.506·15-s + 0.181·17-s − 0.984·19-s + 0.928·21-s + 0.358·23-s + 0.200·25-s + 0.811·27-s − 0.324·29-s + 0.748·31-s − 1.02·33-s − 0.366·35-s + 0.771·37-s + 0.853·39-s − 1.54·41-s + 1.29·43-s + 0.126·45-s + 1.86·47-s − 0.327·49-s − 0.205·51-s − 1.08·53-s + 0.406·55-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 \( 1 + 1.96T + 3T^{2} \)
7 \( 1 + 2.16T + 7T^{2} \)
11 \( 1 - 3.01T + 11T^{2} \)
13 \( 1 + 2.71T + 13T^{2} \)
17 \( 1 - 0.748T + 17T^{2} \)
19 \( 1 + 4.29T + 19T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 + 1.74T + 29T^{2} \)
31 \( 1 - 4.16T + 31T^{2} \)
37 \( 1 - 4.69T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 7.88T + 53T^{2} \)
59 \( 1 - 6.20T + 59T^{2} \)
61 \( 1 + 9.79T + 61T^{2} \)
67 \( 1 - 2.54T + 67T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 + 5.44T + 79T^{2} \)
83 \( 1 - 4.01T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + 2.91T + 97T^{2} \)
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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

−7.48405200601186976402034940732, −6.77002783187613666480984446487, −6.22662497464482199215854512205, −5.79354016972881411883571329411, −4.88635450053561325510248725621, −4.24841509730646980342919338533, −3.20214529425128953804875903106, −2.30612612059409720093110126665, −1.08895334580445830326054490229, 0, 1.08895334580445830326054490229, 2.30612612059409720093110126665, 3.20214529425128953804875903106, 4.24841509730646980342919338533, 4.88635450053561325510248725621, 5.79354016972881411883571329411, 6.22662497464482199215854512205, 6.77002783187613666480984446487, 7.48405200601186976402034940732

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