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  + 2.06·3-s + 5-s − 0.835·7-s + 1.28·9-s − 5.16·11-s + 4.52·13-s + 2.06·15-s − 5.30·17-s − 6.46·19-s − 1.72·21-s − 5.32·23-s + 25-s − 3.55·27-s + 7.21·29-s + 7.66·31-s − 10.6·33-s − 0.835·35-s + 7.65·37-s + 9.35·39-s − 3.14·41-s + 8.02·43-s + 1.28·45-s − 12.1·47-s − 6.30·49-s − 10.9·51-s + 11.3·53-s − 5.16·55-s + ⋯
L(s)  = 1  + 1.19·3-s + 0.447·5-s − 0.315·7-s + 0.427·9-s − 1.55·11-s + 1.25·13-s + 0.534·15-s − 1.28·17-s − 1.48·19-s − 0.377·21-s − 1.11·23-s + 0.200·25-s − 0.684·27-s + 1.33·29-s + 1.37·31-s − 1.86·33-s − 0.141·35-s + 1.25·37-s + 1.49·39-s − 0.490·41-s + 1.22·43-s + 0.191·45-s − 1.77·47-s − 0.900·49-s − 1.53·51-s + 1.55·53-s − 0.696·55-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 \( 1 - 2.06T + 3T^{2} \)
7 \( 1 + 0.835T + 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 - 4.52T + 13T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 + 6.46T + 19T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 - 7.21T + 29T^{2} \)
31 \( 1 - 7.66T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 + 3.14T + 41T^{2} \)
43 \( 1 - 8.02T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 5.34T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + 8.45T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.174050636052027209036941489860, −7.04113390672555096503888427705, −6.24846946040495195557480936079, −5.79867396874907828339826550652, −4.54269116173303139443885787660, −4.08996727175308259083524932827, −2.75477583079264952641076905782, −2.69691138167140702311420846131, −1.63274584579559044498753486014, 0, 1.63274584579559044498753486014, 2.69691138167140702311420846131, 2.75477583079264952641076905782, 4.08996727175308259083524932827, 4.54269116173303139443885787660, 5.79867396874907828339826550652, 6.24846946040495195557480936079, 7.04113390672555096503888427705, 8.174050636052027209036941489860

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