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  − 1.02·3-s + 5-s + 3.96·7-s − 1.94·9-s − 0.133·11-s + 2.05·13-s − 1.02·15-s + 1.43·17-s − 7.44·19-s − 4.07·21-s − 6.51·23-s + 25-s + 5.07·27-s + 3.69·29-s + 6.79·31-s + 0.137·33-s + 3.96·35-s − 10.7·37-s − 2.11·39-s − 11.8·41-s − 3.64·43-s − 1.94·45-s − 6.70·47-s + 8.71·49-s − 1.47·51-s − 11.2·53-s − 0.133·55-s + ⋯
L(s)  = 1  − 0.593·3-s + 0.447·5-s + 1.49·7-s − 0.648·9-s − 0.0403·11-s + 0.569·13-s − 0.265·15-s + 0.348·17-s − 1.70·19-s − 0.888·21-s − 1.35·23-s + 0.200·25-s + 0.977·27-s + 0.686·29-s + 1.21·31-s + 0.0239·33-s + 0.670·35-s − 1.76·37-s − 0.338·39-s − 1.85·41-s − 0.556·43-s − 0.289·45-s − 0.978·47-s + 1.24·49-s − 0.206·51-s − 1.54·53-s − 0.0180·55-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 \( 1 + 1.02T + 3T^{2} \)
7 \( 1 - 3.96T + 7T^{2} \)
11 \( 1 + 0.133T + 11T^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 - 3.69T + 29T^{2} \)
31 \( 1 - 6.79T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 3.64T + 43T^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 0.753T + 59T^{2} \)
61 \( 1 - 3.95T + 61T^{2} \)
67 \( 1 - 0.858T + 67T^{2} \)
71 \( 1 - 2.58T + 71T^{2} \)
73 \( 1 - 8.62T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 6.16T + 89T^{2} \)
97 \( 1 - 0.556T + 97T^{2} \)
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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

−8.051993828143489751061987429683, −6.72792149033121650987370145539, −6.33676013137724004319024741653, −5.50635923232994984192073465638, −4.93540280789369628889807436317, −4.28350593689294888450910784253, −3.20228799327727467703385303559, −2.07434469938763996294955559101, −1.45202256106384352422165990557, 0, 1.45202256106384352422165990557, 2.07434469938763996294955559101, 3.20228799327727467703385303559, 4.28350593689294888450910784253, 4.93540280789369628889807436317, 5.50635923232994984192073465638, 6.33676013137724004319024741653, 6.72792149033121650987370145539, 8.051993828143489751061987429683

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