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  + 0.815·3-s + 5-s − 2.49·7-s − 2.33·9-s + 1.48·11-s + 5.26·13-s + 0.815·15-s + 1.04·17-s − 7.39·19-s − 2.03·21-s + 0.0259·23-s + 25-s − 4.35·27-s + 6.30·29-s − 8.70·31-s + 1.21·33-s − 2.49·35-s + 0.895·37-s + 4.29·39-s + 3.52·41-s − 8.45·43-s − 2.33·45-s − 4.55·47-s − 0.780·49-s + 0.853·51-s − 4.01·53-s + 1.48·55-s + ⋯
L(s)  = 1  + 0.471·3-s + 0.447·5-s − 0.942·7-s − 0.778·9-s + 0.448·11-s + 1.46·13-s + 0.210·15-s + 0.253·17-s − 1.69·19-s − 0.443·21-s + 0.00541·23-s + 0.200·25-s − 0.837·27-s + 1.17·29-s − 1.56·31-s + 0.211·33-s − 0.421·35-s + 0.147·37-s + 0.687·39-s + 0.549·41-s − 1.28·43-s − 0.347·45-s − 0.664·47-s − 0.111·49-s + 0.119·51-s − 0.551·53-s + 0.200·55-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 \( 1 - 0.815T + 3T^{2} \)
7 \( 1 + 2.49T + 7T^{2} \)
11 \( 1 - 1.48T + 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 - 1.04T + 17T^{2} \)
19 \( 1 + 7.39T + 19T^{2} \)
23 \( 1 - 0.0259T + 23T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + 8.70T + 31T^{2} \)
37 \( 1 - 0.895T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 + 8.45T + 43T^{2} \)
47 \( 1 + 4.55T + 47T^{2} \)
53 \( 1 + 4.01T + 53T^{2} \)
59 \( 1 - 4.73T + 59T^{2} \)
61 \( 1 + 8.15T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 + 2.24T + 71T^{2} \)
73 \( 1 + 6.46T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 1.61T + 89T^{2} \)
97 \( 1 + 4.71T + 97T^{2} \)
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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

−7.925625109683067381700015793141, −6.78347326759080363220129573354, −6.26426971290999390842482831675, −5.87650227492028811779842188248, −4.79949075567167048554751466657, −3.76001623275311993737918406126, −3.31354095957875449566759754403, −2.39461507229941037540782507290, −1.44284490147630557330809851261, 0, 1.44284490147630557330809851261, 2.39461507229941037540782507290, 3.31354095957875449566759754403, 3.76001623275311993737918406126, 4.79949075567167048554751466657, 5.87650227492028811779842188248, 6.26426971290999390842482831675, 6.78347326759080363220129573354, 7.925625109683067381700015793141

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