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  + 0.237·3-s + 5-s + 4.27·7-s − 2.94·9-s − 2.02·11-s − 4.81·13-s + 0.237·15-s − 5.46·17-s + 3.56·19-s + 1.01·21-s − 0.164·23-s + 25-s − 1.41·27-s − 7.17·29-s + 8.23·31-s − 0.482·33-s + 4.27·35-s + 7.02·37-s − 1.14·39-s − 7.73·41-s + 3.47·43-s − 2.94·45-s + 7.39·47-s + 11.2·49-s − 1.29·51-s − 5.18·53-s − 2.02·55-s + ⋯
L(s)  = 1  + 0.137·3-s + 0.447·5-s + 1.61·7-s − 0.981·9-s − 0.611·11-s − 1.33·13-s + 0.0613·15-s − 1.32·17-s + 0.816·19-s + 0.221·21-s − 0.0342·23-s + 0.200·25-s − 0.271·27-s − 1.33·29-s + 1.47·31-s − 0.0839·33-s + 0.722·35-s + 1.15·37-s − 0.183·39-s − 1.20·41-s + 0.530·43-s − 0.438·45-s + 1.07·47-s + 1.60·49-s − 0.181·51-s − 0.712·53-s − 0.273·55-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 \( 1 - 0.237T + 3T^{2} \)
7 \( 1 - 4.27T + 7T^{2} \)
11 \( 1 + 2.02T + 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 + 5.46T + 17T^{2} \)
19 \( 1 - 3.56T + 19T^{2} \)
23 \( 1 + 0.164T + 23T^{2} \)
29 \( 1 + 7.17T + 29T^{2} \)
31 \( 1 - 8.23T + 31T^{2} \)
37 \( 1 - 7.02T + 37T^{2} \)
41 \( 1 + 7.73T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + 5.18T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 8.08T + 67T^{2} \)
71 \( 1 + 0.838T + 71T^{2} \)
73 \( 1 + 4.52T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 - 3.11T + 97T^{2} \)
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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

−7.82536554772358394858445414559, −7.21864525592123037949822656582, −6.17993064414701967423981694841, −5.44935106738699999618473957264, −4.86185463543776028074077949801, −4.33172369678776641803377789030, −2.88793725296612992632007083587, −2.39271013303349811930397369439, −1.50265143432741466498284888725, 0, 1.50265143432741466498284888725, 2.39271013303349811930397369439, 2.88793725296612992632007083587, 4.33172369678776641803377789030, 4.86185463543776028074077949801, 5.44935106738699999618473957264, 6.17993064414701967423981694841, 7.21864525592123037949822656582, 7.82536554772358394858445414559

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