Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 79 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 0.642·5-s + 1.33·7-s − 0.996·9-s + 1.93·11-s − 5.21·13-s + 0.909·15-s − 1.65·17-s + 19-s + 1.88·21-s + 1.45·23-s − 4.58·25-s − 5.65·27-s − 0.0964·29-s − 5.73·31-s + 2.73·33-s + 0.855·35-s − 7.59·37-s − 7.38·39-s + 5.41·41-s − 6.60·43-s − 0.640·45-s + 1.50·47-s − 5.22·49-s − 2.33·51-s + 0.0801·53-s + 1.24·55-s + ⋯
L(s)  = 1  + 0.817·3-s + 0.287·5-s + 0.502·7-s − 0.332·9-s + 0.582·11-s − 1.44·13-s + 0.234·15-s − 0.400·17-s + 0.229·19-s + 0.410·21-s + 0.303·23-s − 0.917·25-s − 1.08·27-s − 0.0179·29-s − 1.03·31-s + 0.476·33-s + 0.144·35-s − 1.24·37-s − 1.18·39-s + 0.845·41-s − 1.00·43-s − 0.0954·45-s + 0.219·47-s − 0.747·49-s − 0.327·51-s + 0.0110·53-s + 0.167·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6004,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
79 \( 1 + T \)
good3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 0.642T + 5T^{2} \)
7 \( 1 - 1.33T + 7T^{2} \)
11 \( 1 - 1.93T + 11T^{2} \)
13 \( 1 + 5.21T + 13T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
23 \( 1 - 1.45T + 23T^{2} \)
29 \( 1 + 0.0964T + 29T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
37 \( 1 + 7.59T + 37T^{2} \)
41 \( 1 - 5.41T + 41T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 - 1.50T + 47T^{2} \)
53 \( 1 - 0.0801T + 53T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + 8.01T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
83 \( 1 - 7.39T + 83T^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 + 5.13T + 97T^{2} \)
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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

−7.69099976771817310548183997684, −7.25113662570877889221235576034, −6.35391930496990045922709303362, −5.47588318543766227656594815739, −4.86467877544302717520802900291, −3.95059619323611404099457744969, −3.15057560079854314075267421628, −2.28762072473784424465808238226, −1.65228235002582736535200043720, 0, 1.65228235002582736535200043720, 2.28762072473784424465808238226, 3.15057560079854314075267421628, 3.95059619323611404099457744969, 4.86467877544302717520802900291, 5.47588318543766227656594815739, 6.35391930496990045922709303362, 7.25113662570877889221235576034, 7.69099976771817310548183997684

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