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  − 2.44·3-s + 5-s − 4.90·7-s + 2.97·9-s − 4.55·11-s + 3.11·13-s − 2.44·15-s − 4.54·17-s − 0.659·19-s + 11.9·21-s + 9.23·23-s + 25-s + 0.0713·27-s + 6.59·29-s − 8.71·31-s + 11.1·33-s − 4.90·35-s + 1.00·37-s − 7.62·39-s − 6.56·41-s + 4.24·43-s + 2.97·45-s + 5.91·47-s + 17.0·49-s + 11.1·51-s + 5.36·53-s − 4.55·55-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.447·5-s − 1.85·7-s + 0.990·9-s − 1.37·11-s + 0.865·13-s − 0.630·15-s − 1.10·17-s − 0.151·19-s + 2.61·21-s + 1.92·23-s + 0.200·25-s + 0.0137·27-s + 1.22·29-s − 1.56·31-s + 1.93·33-s − 0.829·35-s + 0.165·37-s − 1.22·39-s − 1.02·41-s + 0.647·43-s + 0.442·45-s + 0.863·47-s + 2.43·49-s + 1.55·51-s + 0.737·53-s − 0.614·55-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 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 + 4.90T + 7T^{2} \)
11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 + 4.54T + 17T^{2} \)
19 \( 1 + 0.659T + 19T^{2} \)
23 \( 1 - 9.23T + 23T^{2} \)
29 \( 1 - 6.59T + 29T^{2} \)
31 \( 1 + 8.71T + 31T^{2} \)
37 \( 1 - 1.00T + 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 - 5.91T + 47T^{2} \)
53 \( 1 - 5.36T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 2.04T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 7.82T + 71T^{2} \)
73 \( 1 - 5.78T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 3.88T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 - 0.995T + 97T^{2} \)
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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

−7.34214461066769593416011993203, −6.73574006314035104768236277012, −6.36205990909484382880494519869, −5.55207040783403873957535082391, −5.18960810924101566664058397113, −4.14693959782672172310147793955, −3.14407698905030335979847577830, −2.45195845603101473367589379587, −0.898756443363507894668429997423, 0, 0.898756443363507894668429997423, 2.45195845603101473367589379587, 3.14407698905030335979847577830, 4.14693959782672172310147793955, 5.18960810924101566664058397113, 5.55207040783403873957535082391, 6.36205990909484382880494519869, 6.73574006314035104768236277012, 7.34214461066769593416011993203

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