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  − 1.90·3-s + 5-s + 1.15·7-s + 0.640·9-s + 4.77·11-s + 3.49·13-s − 1.90·15-s − 7.78·17-s + 1.91·19-s − 2.19·21-s − 6.30·23-s + 25-s + 4.50·27-s − 0.185·29-s − 9.77·31-s − 9.10·33-s + 1.15·35-s + 9.42·37-s − 6.66·39-s − 3.34·41-s − 1.84·43-s + 0.640·45-s − 1.21·47-s − 5.67·49-s + 14.8·51-s − 0.503·53-s + 4.77·55-s + ⋯
L(s)  = 1  − 1.10·3-s + 0.447·5-s + 0.435·7-s + 0.213·9-s + 1.43·11-s + 0.968·13-s − 0.492·15-s − 1.88·17-s + 0.438·19-s − 0.479·21-s − 1.31·23-s + 0.200·25-s + 0.866·27-s − 0.0344·29-s − 1.75·31-s − 1.58·33-s + 0.194·35-s + 1.54·37-s − 1.06·39-s − 0.521·41-s − 0.281·43-s + 0.0955·45-s − 0.176·47-s − 0.810·49-s + 2.07·51-s − 0.0691·53-s + 0.643·55-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 \( 1 + 1.90T + 3T^{2} \)
7 \( 1 - 1.15T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 + 7.78T + 17T^{2} \)
19 \( 1 - 1.91T + 19T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 + 0.185T + 29T^{2} \)
31 \( 1 + 9.77T + 31T^{2} \)
37 \( 1 - 9.42T + 37T^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 + 1.84T + 43T^{2} \)
47 \( 1 + 1.21T + 47T^{2} \)
53 \( 1 + 0.503T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 7.19T + 61T^{2} \)
67 \( 1 - 1.44T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 5.53T + 73T^{2} \)
79 \( 1 + 8.71T + 79T^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 - 1.98T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−7.60509814088554676604567789787, −6.73165951067572401589292787552, −6.12732423503704492009228398643, −5.89790281945093730876620716427, −4.78276934506531315006402302107, −4.28113230568115047551785113536, −3.36136209545697100809321416226, −2.01300154652783139825887234388, −1.32159077462744440224022369779, 0, 1.32159077462744440224022369779, 2.01300154652783139825887234388, 3.36136209545697100809321416226, 4.28113230568115047551785113536, 4.78276934506531315006402302107, 5.89790281945093730876620716427, 6.12732423503704492009228398643, 6.73165951067572401589292787552, 7.60509814088554676604567789787

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