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  + 1.56·3-s − 2.37·5-s + 2.65·7-s − 0.557·9-s − 2.81·11-s + 2.78·13-s − 3.71·15-s − 7.35·17-s + 19-s + 4.15·21-s + 5.90·23-s + 0.642·25-s − 5.55·27-s + 2.21·29-s + 1.03·31-s − 4.39·33-s − 6.31·35-s + 4.56·37-s + 4.36·39-s + 2.65·41-s + 6.23·43-s + 1.32·45-s + 0.283·47-s + 0.0658·49-s − 11.4·51-s − 8.74·53-s + 6.68·55-s + ⋯
L(s)  = 1  + 0.902·3-s − 1.06·5-s + 1.00·7-s − 0.185·9-s − 0.847·11-s + 0.773·13-s − 0.958·15-s − 1.78·17-s + 0.229·19-s + 0.906·21-s + 1.23·23-s + 0.128·25-s − 1.07·27-s + 0.411·29-s + 0.185·31-s − 0.765·33-s − 1.06·35-s + 0.750·37-s + 0.698·39-s + 0.414·41-s + 0.951·43-s + 0.197·45-s + 0.0413·47-s + 0.00940·49-s − 1.60·51-s − 1.20·53-s + 0.900·55-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 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 + 2.37T + 5T^{2} \)
7 \( 1 - 2.65T + 7T^{2} \)
11 \( 1 + 2.81T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 + 7.35T + 17T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 - 2.21T + 29T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 - 4.56T + 37T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 - 0.283T + 47T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 2.64T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 0.602T + 71T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.86476020214158626292067026843, −7.34835949806269485757130555346, −6.41386792425512732114392676301, −5.47068311688547321295190019084, −4.54958116295088684230550486657, −4.15624928206801817502538946543, −3.07997383879319506901666839838, −2.54816457309780041194297468909, −1.42853497338071704016563511861, 0, 1.42853497338071704016563511861, 2.54816457309780041194297468909, 3.07997383879319506901666839838, 4.15624928206801817502538946543, 4.54958116295088684230550486657, 5.47068311688547321295190019084, 6.41386792425512732114392676301, 7.34835949806269485757130555346, 7.86476020214158626292067026843

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