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  − 2.89·3-s + 5-s + 0.451·7-s + 5.36·9-s − 4.25·11-s − 4.52·13-s − 2.89·15-s + 2.97·17-s − 0.416·19-s − 1.30·21-s + 4.36·23-s + 25-s − 6.84·27-s − 1.71·29-s + 0.519·31-s + 12.3·33-s + 0.451·35-s + 5.90·37-s + 13.0·39-s + 7.30·41-s + 6.72·43-s + 5.36·45-s − 7.34·47-s − 6.79·49-s − 8.60·51-s − 8.53·53-s − 4.25·55-s + ⋯
L(s)  = 1  − 1.66·3-s + 0.447·5-s + 0.170·7-s + 1.78·9-s − 1.28·11-s − 1.25·13-s − 0.746·15-s + 0.721·17-s − 0.0955·19-s − 0.284·21-s + 0.910·23-s + 0.200·25-s − 1.31·27-s − 0.318·29-s + 0.0932·31-s + 2.14·33-s + 0.0762·35-s + 0.971·37-s + 2.09·39-s + 1.14·41-s + 1.02·43-s + 0.799·45-s − 1.07·47-s − 0.970·49-s − 1.20·51-s − 1.17·53-s − 0.573·55-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 \( 1 + 2.89T + 3T^{2} \)
7 \( 1 - 0.451T + 7T^{2} \)
11 \( 1 + 4.25T + 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 - 2.97T + 17T^{2} \)
19 \( 1 + 0.416T + 19T^{2} \)
23 \( 1 - 4.36T + 23T^{2} \)
29 \( 1 + 1.71T + 29T^{2} \)
31 \( 1 - 0.519T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 - 7.30T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 + 8.53T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 3.77T + 67T^{2} \)
71 \( 1 - 9.96T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 2.27T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−7.52990465899953410354244284908, −6.97827500417358057047430758977, −6.06027581866459434158160302243, −5.62203125917150596288131320366, −4.87051896255424216260017970092, −4.60099048680208281570902220352, −3.13149173451312533693668381102, −2.22673491102142053037020735590, −1.03833463078192215057342192094, 0, 1.03833463078192215057342192094, 2.22673491102142053037020735590, 3.13149173451312533693668381102, 4.60099048680208281570902220352, 4.87051896255424216260017970092, 5.62203125917150596288131320366, 6.06027581866459434158160302243, 6.97827500417358057047430758977, 7.52990465899953410354244284908

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