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  − 1.68·3-s + 5-s − 4.78·7-s − 0.157·9-s − 1.17·11-s + 3.55·13-s − 1.68·15-s + 5.24·17-s + 2.24·19-s + 8.07·21-s − 8.41·23-s + 25-s + 5.32·27-s − 8.71·29-s + 3.38·31-s + 1.98·33-s − 4.78·35-s + 9.53·37-s − 5.98·39-s − 2.73·41-s + 4.70·43-s − 0.157·45-s − 11.7·47-s + 15.9·49-s − 8.83·51-s − 4.56·53-s − 1.17·55-s + ⋯
L(s)  = 1  − 0.973·3-s + 0.447·5-s − 1.81·7-s − 0.0526·9-s − 0.355·11-s + 0.984·13-s − 0.435·15-s + 1.27·17-s + 0.515·19-s + 1.76·21-s − 1.75·23-s + 0.200·25-s + 1.02·27-s − 1.61·29-s + 0.608·31-s + 0.346·33-s − 0.809·35-s + 1.56·37-s − 0.958·39-s − 0.427·41-s + 0.718·43-s − 0.0235·45-s − 1.70·47-s + 2.27·49-s − 1.23·51-s − 0.626·53-s − 0.158·55-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 \( 1 + 1.68T + 3T^{2} \)
7 \( 1 + 4.78T + 7T^{2} \)
11 \( 1 + 1.17T + 11T^{2} \)
13 \( 1 - 3.55T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 8.41T + 23T^{2} \)
29 \( 1 + 8.71T + 29T^{2} \)
31 \( 1 - 3.38T + 31T^{2} \)
37 \( 1 - 9.53T + 37T^{2} \)
41 \( 1 + 2.73T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 4.56T + 53T^{2} \)
59 \( 1 - 2.95T + 59T^{2} \)
61 \( 1 - 3.98T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + 3.61T + 71T^{2} \)
73 \( 1 - 9.48T + 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 0.962T + 89T^{2} \)
97 \( 1 - 3.59T + 97T^{2} \)
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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

−7.70280953941152396887441647764, −6.66970960092849035350591756671, −6.16061641758384371620806799827, −5.80250923931541904504590130203, −5.15154693883881559372508898311, −3.85602601733083868207878617878, −3.35175661579419837652679891517, −2.39838588318668420463097765486, −1.02611236756747229705119350687, 0, 1.02611236756747229705119350687, 2.39838588318668420463097765486, 3.35175661579419837652679891517, 3.85602601733083868207878617878, 5.15154693883881559372508898311, 5.80250923931541904504590130203, 6.16061641758384371620806799827, 6.66970960092849035350591756671, 7.70280953941152396887441647764

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