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  − 0.907·3-s + 1.31·5-s − 2.29·7-s − 2.17·9-s + 0.212·11-s + 6.06·13-s − 1.19·15-s + 0.799·17-s + 19-s + 2.08·21-s + 0.602·23-s − 3.26·25-s + 4.69·27-s − 7.49·29-s − 1.42·31-s − 0.193·33-s − 3.02·35-s − 8.32·37-s − 5.50·39-s + 0.745·41-s + 8.77·43-s − 2.86·45-s − 9.64·47-s − 1.73·49-s − 0.725·51-s − 9.32·53-s + 0.279·55-s + ⋯
L(s)  = 1  − 0.524·3-s + 0.588·5-s − 0.867·7-s − 0.725·9-s + 0.0641·11-s + 1.68·13-s − 0.308·15-s + 0.193·17-s + 0.229·19-s + 0.454·21-s + 0.125·23-s − 0.653·25-s + 0.904·27-s − 1.39·29-s − 0.256·31-s − 0.0336·33-s − 0.510·35-s − 1.36·37-s − 0.881·39-s + 0.116·41-s + 1.33·43-s − 0.427·45-s − 1.40·47-s − 0.247·49-s − 0.101·51-s − 1.28·53-s + 0.0377·55-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 \( 1 + 0.907T + 3T^{2} \)
5 \( 1 - 1.31T + 5T^{2} \)
7 \( 1 + 2.29T + 7T^{2} \)
11 \( 1 - 0.212T + 11T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 - 0.799T + 17T^{2} \)
23 \( 1 - 0.602T + 23T^{2} \)
29 \( 1 + 7.49T + 29T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 + 8.32T + 37T^{2} \)
41 \( 1 - 0.745T + 41T^{2} \)
43 \( 1 - 8.77T + 43T^{2} \)
47 \( 1 + 9.64T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 2.17T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 6.30T + 73T^{2} \)
83 \( 1 - 8.03T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 7.60T + 97T^{2} \)
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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

−7.77612174399329224186090741558, −6.66999902125881496667037500895, −6.31882292600919014246570397346, −5.63115723898881339458264788973, −5.16707960715525535140385145414, −3.77467889330193774228775022164, −3.42534861178414988101244388477, −2.29431320169549548044986525612, −1.24670349009693493441986806405, 0, 1.24670349009693493441986806405, 2.29431320169549548044986525612, 3.42534861178414988101244388477, 3.77467889330193774228775022164, 5.16707960715525535140385145414, 5.63115723898881339458264788973, 6.31882292600919014246570397346, 6.66999902125881496667037500895, 7.77612174399329224186090741558

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