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  + 2.35·3-s + 5-s + 0.202·7-s + 2.54·9-s − 4.49·11-s − 0.979·13-s + 2.35·15-s + 0.563·17-s − 3.51·19-s + 0.477·21-s − 8.46·23-s + 25-s − 1.06·27-s − 3.31·29-s + 5.73·31-s − 10.5·33-s + 0.202·35-s − 8.46·37-s − 2.30·39-s − 6.59·41-s − 0.444·43-s + 2.54·45-s + 10.8·47-s − 6.95·49-s + 1.32·51-s − 6.63·53-s − 4.49·55-s + ⋯
L(s)  = 1  + 1.35·3-s + 0.447·5-s + 0.0766·7-s + 0.848·9-s − 1.35·11-s − 0.271·13-s + 0.608·15-s + 0.136·17-s − 0.806·19-s + 0.104·21-s − 1.76·23-s + 0.200·25-s − 0.205·27-s − 0.615·29-s + 1.02·31-s − 1.84·33-s + 0.0342·35-s − 1.39·37-s − 0.369·39-s − 1.03·41-s − 0.0677·43-s + 0.379·45-s + 1.58·47-s − 0.994·49-s + 0.185·51-s − 0.911·53-s − 0.605·55-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 \( 1 - 2.35T + 3T^{2} \)
7 \( 1 - 0.202T + 7T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 + 0.979T + 13T^{2} \)
17 \( 1 - 0.563T + 17T^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 + 8.46T + 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 + 8.46T + 37T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 + 0.444T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 6.63T + 53T^{2} \)
59 \( 1 - 1.09T + 59T^{2} \)
61 \( 1 - 0.633T + 61T^{2} \)
67 \( 1 - 4.51T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 0.361T + 73T^{2} \)
79 \( 1 - 2.63T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 6.68T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.961562400288021234623811863737, −7.25376270490065603437601703566, −6.32404204073243482935269584198, −5.55822224881183529122459783187, −4.76341958253666961918550738151, −3.87616085428272897808057328073, −3.08319142522861840056536955649, −2.31609462998700022563528253407, −1.80291184482599512488421391609, 0, 1.80291184482599512488421391609, 2.31609462998700022563528253407, 3.08319142522861840056536955649, 3.87616085428272897808057328073, 4.76341958253666961918550738151, 5.55822224881183529122459783187, 6.32404204073243482935269584198, 7.25376270490065603437601703566, 7.961562400288021234623811863737

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