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  − 3.33·3-s + 5-s + 2.35·7-s + 8.13·9-s − 5.41·11-s + 7.10·13-s − 3.33·15-s − 2.30·17-s + 3.33·19-s − 7.85·21-s − 2.59·23-s + 25-s − 17.1·27-s − 1.45·29-s + 2.14·31-s + 18.0·33-s + 2.35·35-s − 8.97·37-s − 23.7·39-s + 0.901·41-s − 1.97·43-s + 8.13·45-s − 11.7·47-s − 1.46·49-s + 7.69·51-s + 10.0·53-s − 5.41·55-s + ⋯
L(s)  = 1  − 1.92·3-s + 0.447·5-s + 0.889·7-s + 2.71·9-s − 1.63·11-s + 1.97·13-s − 0.861·15-s − 0.559·17-s + 0.765·19-s − 1.71·21-s − 0.541·23-s + 0.200·25-s − 3.29·27-s − 0.271·29-s + 0.385·31-s + 3.14·33-s + 0.397·35-s − 1.47·37-s − 3.79·39-s + 0.140·41-s − 0.301·43-s + 1.21·45-s − 1.71·47-s − 0.209·49-s + 1.07·51-s + 1.37·53-s − 0.729·55-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 \( 1 + 3.33T + 3T^{2} \)
7 \( 1 - 2.35T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 7.10T + 13T^{2} \)
17 \( 1 + 2.30T + 17T^{2} \)
19 \( 1 - 3.33T + 19T^{2} \)
23 \( 1 + 2.59T + 23T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 + 8.97T + 37T^{2} \)
41 \( 1 - 0.901T + 41T^{2} \)
43 \( 1 + 1.97T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 4.21T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 0.385T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 6.72T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 2.69T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 5.95T + 97T^{2} \)
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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

−7.66065631983843820231351750607, −6.77580002105806219226991707983, −6.12814002794764713683714138074, −5.55928411265913408147291036363, −5.05640380484517641352097661317, −4.42547314409810870690633108874, −3.37255595062065185746513183012, −1.89399004432108335901745227681, −1.21891355678433128626333278294, 0, 1.21891355678433128626333278294, 1.89399004432108335901745227681, 3.37255595062065185746513183012, 4.42547314409810870690633108874, 5.05640380484517641352097661317, 5.55928411265913408147291036363, 6.12814002794764713683714138074, 6.77580002105806219226991707983, 7.66065631983843820231351750607

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