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  + 2.11·3-s + 5-s − 4.51·7-s + 1.47·9-s + 3.71·11-s + 2.26·13-s + 2.11·15-s − 7.69·17-s − 3.28·19-s − 9.54·21-s + 3.96·23-s + 25-s − 3.22·27-s − 10.2·29-s + 7.05·31-s + 7.87·33-s − 4.51·35-s − 2.55·37-s + 4.79·39-s − 1.38·41-s + 2.98·43-s + 1.47·45-s + 1.63·47-s + 13.3·49-s − 16.2·51-s − 11.1·53-s + 3.71·55-s + ⋯
L(s)  = 1  + 1.22·3-s + 0.447·5-s − 1.70·7-s + 0.492·9-s + 1.12·11-s + 0.629·13-s + 0.546·15-s − 1.86·17-s − 0.754·19-s − 2.08·21-s + 0.826·23-s + 0.200·25-s − 0.620·27-s − 1.89·29-s + 1.26·31-s + 1.37·33-s − 0.762·35-s − 0.419·37-s + 0.768·39-s − 0.216·41-s + 0.454·43-s + 0.220·45-s + 0.238·47-s + 1.90·49-s − 2.28·51-s − 1.53·53-s + 0.501·55-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 \( 1 - 2.11T + 3T^{2} \)
7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + 7.69T + 17T^{2} \)
19 \( 1 + 3.28T + 19T^{2} \)
23 \( 1 - 3.96T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 7.05T + 31T^{2} \)
37 \( 1 + 2.55T + 37T^{2} \)
41 \( 1 + 1.38T + 41T^{2} \)
43 \( 1 - 2.98T + 43T^{2} \)
47 \( 1 - 1.63T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 8.02T + 59T^{2} \)
61 \( 1 + 8.37T + 61T^{2} \)
67 \( 1 - 2.46T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 7.06T + 73T^{2} \)
79 \( 1 + 7.66T + 79T^{2} \)
83 \( 1 + 6.35T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 - 5.64T + 97T^{2} \)
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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

−7.76575771736876052798178762999, −6.85788439036636779816606716343, −6.42660434705618843864414600567, −5.90409635304444489276582902091, −4.52846368299052849531475265085, −3.82502966417887838060184596392, −3.19306245180642580388496701035, −2.47790204808043551171845510086, −1.59711316806124094202263206183, 0, 1.59711316806124094202263206183, 2.47790204808043551171845510086, 3.19306245180642580388496701035, 3.82502966417887838060184596392, 4.52846368299052849531475265085, 5.90409635304444489276582902091, 6.42660434705618843864414600567, 6.85788439036636779816606716343, 7.76575771736876052798178762999

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