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  + 1.21·3-s + 1.36·5-s − 2.61·7-s − 1.51·9-s − 3.78·11-s + 1.85·13-s + 1.66·15-s + 2.70·17-s + 19-s − 3.19·21-s + 7.26·23-s − 3.14·25-s − 5.50·27-s + 1.91·29-s + 5.73·31-s − 4.61·33-s − 3.56·35-s − 4.70·37-s + 2.26·39-s − 10.8·41-s − 12.3·43-s − 2.06·45-s + 8.24·47-s − 0.140·49-s + 3.29·51-s + 8.29·53-s − 5.15·55-s + ⋯
L(s)  = 1  + 0.704·3-s + 0.609·5-s − 0.989·7-s − 0.504·9-s − 1.14·11-s + 0.515·13-s + 0.428·15-s + 0.655·17-s + 0.229·19-s − 0.696·21-s + 1.51·23-s − 0.628·25-s − 1.05·27-s + 0.355·29-s + 1.03·31-s − 0.803·33-s − 0.602·35-s − 0.773·37-s + 0.362·39-s − 1.69·41-s − 1.88·43-s − 0.307·45-s + 1.20·47-s − 0.0201·49-s + 0.461·51-s + 1.13·53-s − 0.694·55-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 \( 1 - 1.21T + 3T^{2} \)
5 \( 1 - 1.36T + 5T^{2} \)
7 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 + 3.78T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
23 \( 1 - 7.26T + 23T^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 - 8.29T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 0.427T + 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 + 0.159T + 71T^{2} \)
73 \( 1 - 2.28T + 73T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−7.88003623975493926441733922622, −6.97911691201919237159911090718, −6.35258369813654499411197460563, −5.50935157936161372811525504072, −5.06705691484587601157284883875, −3.73952466242730246747362810977, −3.04478091434079635008546473945, −2.63046451108006911869480519273, −1.44798197986885863401234133857, 0, 1.44798197986885863401234133857, 2.63046451108006911869480519273, 3.04478091434079635008546473945, 3.73952466242730246747362810977, 5.06705691484587601157284883875, 5.50935157936161372811525504072, 6.35258369813654499411197460563, 6.97911691201919237159911090718, 7.88003623975493926441733922622

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