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  + 3.37·3-s + 5-s − 3.80·7-s + 8.37·9-s − 2.57·11-s − 4.49·13-s + 3.37·15-s + 4.43·17-s − 6.71·19-s − 12.8·21-s − 8.16·23-s + 25-s + 18.1·27-s − 8.65·29-s − 2.63·31-s − 8.69·33-s − 3.80·35-s − 2.00·37-s − 15.1·39-s + 0.367·41-s − 4.23·43-s + 8.37·45-s − 6.96·47-s + 7.48·49-s + 14.9·51-s + 12.5·53-s − 2.57·55-s + ⋯
L(s)  = 1  + 1.94·3-s + 0.447·5-s − 1.43·7-s + 2.79·9-s − 0.776·11-s − 1.24·13-s + 0.870·15-s + 1.07·17-s − 1.54·19-s − 2.80·21-s − 1.70·23-s + 0.200·25-s + 3.49·27-s − 1.60·29-s − 0.473·31-s − 1.51·33-s − 0.643·35-s − 0.329·37-s − 2.42·39-s + 0.0573·41-s − 0.645·43-s + 1.24·45-s − 1.01·47-s + 1.06·49-s + 2.09·51-s + 1.72·53-s − 0.347·55-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 \( 1 - 3.37T + 3T^{2} \)
7 \( 1 + 3.80T + 7T^{2} \)
11 \( 1 + 2.57T + 11T^{2} \)
13 \( 1 + 4.49T + 13T^{2} \)
17 \( 1 - 4.43T + 17T^{2} \)
19 \( 1 + 6.71T + 19T^{2} \)
23 \( 1 + 8.16T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + 2.00T + 37T^{2} \)
41 \( 1 - 0.367T + 41T^{2} \)
43 \( 1 + 4.23T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 2.01T + 59T^{2} \)
61 \( 1 + 6.13T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 3.21T + 71T^{2} \)
73 \( 1 - 0.138T + 73T^{2} \)
79 \( 1 - 6.47T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 - 7.03T + 97T^{2} \)
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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

−7.72701819504685875985344803479, −7.29047167571958246352272299530, −6.47568022283739267404125478169, −5.64959225465095524149803297128, −4.56803943322035457059269495770, −3.70800359582127848818096898664, −3.19780283020382314112487015400, −2.33041584617927490029306723576, −1.91314137230346783544968366268, 0, 1.91314137230346783544968366268, 2.33041584617927490029306723576, 3.19780283020382314112487015400, 3.70800359582127848818096898664, 4.56803943322035457059269495770, 5.64959225465095524149803297128, 6.47568022283739267404125478169, 7.29047167571958246352272299530, 7.72701819504685875985344803479

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