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  + 0.435·3-s + 5-s − 2.87·7-s − 2.81·9-s − 2.29·11-s − 0.0953·13-s + 0.435·15-s + 3.92·17-s + 6.59·19-s − 1.25·21-s − 0.150·23-s + 25-s − 2.53·27-s + 2.49·29-s − 0.535·31-s − 1.00·33-s − 2.87·35-s − 11.7·37-s − 0.0415·39-s + 9.05·41-s + 8.32·43-s − 2.81·45-s − 1.88·47-s + 1.28·49-s + 1.70·51-s + 10.4·53-s − 2.29·55-s + ⋯
L(s)  = 1  + 0.251·3-s + 0.447·5-s − 1.08·7-s − 0.936·9-s − 0.692·11-s − 0.0264·13-s + 0.112·15-s + 0.951·17-s + 1.51·19-s − 0.273·21-s − 0.0313·23-s + 0.200·25-s − 0.487·27-s + 0.464·29-s − 0.0961·31-s − 0.174·33-s − 0.486·35-s − 1.93·37-s − 0.00664·39-s + 1.41·41-s + 1.26·43-s − 0.418·45-s − 0.274·47-s + 0.184·49-s + 0.239·51-s + 1.42·53-s − 0.309·55-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 \( 1 - 0.435T + 3T^{2} \)
7 \( 1 + 2.87T + 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 + 0.0953T + 13T^{2} \)
17 \( 1 - 3.92T + 17T^{2} \)
19 \( 1 - 6.59T + 19T^{2} \)
23 \( 1 + 0.150T + 23T^{2} \)
29 \( 1 - 2.49T + 29T^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 - 9.05T + 41T^{2} \)
43 \( 1 - 8.32T + 43T^{2} \)
47 \( 1 + 1.88T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 2.46T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 4.05T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + 8.49T + 83T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.59607655436493967387305959244, −7.18033561033395158671778157360, −6.08252642699696339666069114007, −5.69728467221211843213540883357, −5.04448065750752537969112394916, −3.82619987244305728036349684587, −2.98678416882571954384076988320, −2.69655819426724516971941966233, −1.29168242660714256757777483219, 0, 1.29168242660714256757777483219, 2.69655819426724516971941966233, 2.98678416882571954384076988320, 3.82619987244305728036349684587, 5.04448065750752537969112394916, 5.69728467221211843213540883357, 6.08252642699696339666069114007, 7.18033561033395158671778157360, 7.59607655436493967387305959244

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