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  − 1.13·3-s + 5-s + 2.44·7-s − 1.72·9-s + 3.35·11-s − 0.198·13-s − 1.13·15-s + 1.66·17-s − 2.14·19-s − 2.76·21-s − 7.53·23-s + 25-s + 5.33·27-s − 5.30·29-s − 3.84·31-s − 3.79·33-s + 2.44·35-s + 4.72·37-s + 0.225·39-s + 0.116·41-s − 10.5·43-s − 1.72·45-s − 1.58·47-s − 1.04·49-s − 1.87·51-s − 8.98·53-s + 3.35·55-s + ⋯
L(s)  = 1  − 0.652·3-s + 0.447·5-s + 0.922·7-s − 0.573·9-s + 1.01·11-s − 0.0551·13-s − 0.291·15-s + 0.402·17-s − 0.491·19-s − 0.602·21-s − 1.57·23-s + 0.200·25-s + 1.02·27-s − 0.985·29-s − 0.691·31-s − 0.660·33-s + 0.412·35-s + 0.777·37-s + 0.0360·39-s + 0.0181·41-s − 1.61·43-s − 0.256·45-s − 0.230·47-s − 0.148·49-s − 0.263·51-s − 1.23·53-s + 0.452·55-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 \( 1 + 1.13T + 3T^{2} \)
7 \( 1 - 2.44T + 7T^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 + 0.198T + 13T^{2} \)
17 \( 1 - 1.66T + 17T^{2} \)
19 \( 1 + 2.14T + 19T^{2} \)
23 \( 1 + 7.53T + 23T^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
31 \( 1 + 3.84T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 - 0.116T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 1.58T + 47T^{2} \)
53 \( 1 + 8.98T + 53T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 + 5.10T + 61T^{2} \)
67 \( 1 + 2.74T + 67T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 - 5.89T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 8.51T + 97T^{2} \)
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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

−7.916641821665851397992489823653, −6.81301785222138965031309764771, −6.21330559003673573133719584373, −5.63884533963285313642586064572, −4.93875001096632055728814255324, −4.17151105377387771693280307566, −3.28894979636890304595798873552, −2.07063487157153659723220764878, −1.41276329652155994689393060864, 0, 1.41276329652155994689393060864, 2.07063487157153659723220764878, 3.28894979636890304595798873552, 4.17151105377387771693280307566, 4.93875001096632055728814255324, 5.63884533963285313642586064572, 6.21330559003673573133719584373, 6.81301785222138965031309764771, 7.916641821665851397992489823653

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