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  + 1.97·3-s + 5-s − 0.188·7-s + 0.892·9-s + 1.45·11-s − 5.70·13-s + 1.97·15-s − 3.29·17-s − 2.35·19-s − 0.371·21-s − 0.811·23-s + 25-s − 4.15·27-s + 6.35·29-s − 1.00·31-s + 2.87·33-s − 0.188·35-s − 6.49·37-s − 11.2·39-s − 9.03·41-s − 10.4·43-s + 0.892·45-s + 7.03·47-s − 6.96·49-s − 6.50·51-s + 5.21·53-s + 1.45·55-s + ⋯
L(s)  = 1  + 1.13·3-s + 0.447·5-s − 0.0712·7-s + 0.297·9-s + 0.439·11-s − 1.58·13-s + 0.509·15-s − 0.799·17-s − 0.540·19-s − 0.0811·21-s − 0.169·23-s + 0.200·25-s − 0.800·27-s + 1.17·29-s − 0.180·31-s + 0.500·33-s − 0.0318·35-s − 1.06·37-s − 1.80·39-s − 1.41·41-s − 1.58·43-s + 0.133·45-s + 1.02·47-s − 0.994·49-s − 0.910·51-s + 0.716·53-s + 0.196·55-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 \( 1 - 1.97T + 3T^{2} \)
7 \( 1 + 0.188T + 7T^{2} \)
11 \( 1 - 1.45T + 11T^{2} \)
13 \( 1 + 5.70T + 13T^{2} \)
17 \( 1 + 3.29T + 17T^{2} \)
19 \( 1 + 2.35T + 19T^{2} \)
23 \( 1 + 0.811T + 23T^{2} \)
29 \( 1 - 6.35T + 29T^{2} \)
31 \( 1 + 1.00T + 31T^{2} \)
37 \( 1 + 6.49T + 37T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 7.03T + 47T^{2} \)
53 \( 1 - 5.21T + 53T^{2} \)
59 \( 1 - 7.83T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 0.795T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 9.53T + 73T^{2} \)
79 \( 1 + 17.1T + 79T^{2} \)
83 \( 1 - 6.51T + 83T^{2} \)
89 \( 1 + 7.05T + 89T^{2} \)
97 \( 1 - 5.93T + 97T^{2} \)
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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

−7.893756437930285092045074963174, −6.89100654912665959790232653930, −6.64617814194543569828441668986, −5.45099224154532415426252474582, −4.79176067029622717623813689390, −3.94007451606938330453389271232, −3.05271459057623122443468133851, −2.37492913390105938249132819726, −1.70393370706114242807146674208, 0, 1.70393370706114242807146674208, 2.37492913390105938249132819726, 3.05271459057623122443468133851, 3.94007451606938330453389271232, 4.79176067029622717623813689390, 5.45099224154532415426252474582, 6.64617814194543569828441668986, 6.89100654912665959790232653930, 7.893756437930285092045074963174

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