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  − 0.338·3-s + 0.413·5-s − 1.61·7-s − 2.88·9-s + 1.62·11-s + 1.42·13-s − 0.140·15-s + 0.598·17-s + 19-s + 0.545·21-s − 0.815·23-s − 4.82·25-s + 1.99·27-s + 7.29·29-s − 4.96·31-s − 0.549·33-s − 0.667·35-s + 5.22·37-s − 0.481·39-s + 2.10·41-s − 5.53·43-s − 1.19·45-s + 5.12·47-s − 4.40·49-s − 0.202·51-s − 5.32·53-s + 0.672·55-s + ⋯
L(s)  = 1  − 0.195·3-s + 0.185·5-s − 0.609·7-s − 0.961·9-s + 0.489·11-s + 0.394·13-s − 0.0361·15-s + 0.145·17-s + 0.229·19-s + 0.118·21-s − 0.170·23-s − 0.965·25-s + 0.383·27-s + 1.35·29-s − 0.891·31-s − 0.0956·33-s − 0.112·35-s + 0.859·37-s − 0.0770·39-s + 0.329·41-s − 0.843·43-s − 0.178·45-s + 0.747·47-s − 0.629·49-s − 0.0283·51-s − 0.731·53-s + 0.0906·55-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 \( 1 + 0.338T + 3T^{2} \)
5 \( 1 - 0.413T + 5T^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 - 1.42T + 13T^{2} \)
17 \( 1 - 0.598T + 17T^{2} \)
23 \( 1 + 0.815T + 23T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 - 5.22T + 37T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
43 \( 1 + 5.53T + 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 + 5.32T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 6.82T + 73T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 0.614T + 89T^{2} \)
97 \( 1 + 2.40T + 97T^{2} \)
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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

−7.79916891783509350130993731148, −6.87461945935253747666519334598, −6.19885022889159293957892850301, −5.77506879673873772344217729896, −4.90956122100070754334533808853, −3.95636639231423588036277958947, −3.22482419154965977360306538323, −2.42683459748559173410240262332, −1.24582298967227430404226807957, 0, 1.24582298967227430404226807957, 2.42683459748559173410240262332, 3.22482419154965977360306538323, 3.95636639231423588036277958947, 4.90956122100070754334533808853, 5.77506879673873772344217729896, 6.19885022889159293957892850301, 6.87461945935253747666519334598, 7.79916891783509350130993731148

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