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  − 2.14·3-s + 2.12·5-s − 4.35·7-s + 1.59·9-s + 2.62·11-s + 1.60·13-s − 4.55·15-s − 7.43·17-s + 19-s + 9.34·21-s + 6.34·23-s − 0.481·25-s + 3.00·27-s − 6.78·29-s + 5.19·31-s − 5.63·33-s − 9.26·35-s + 9.25·37-s − 3.44·39-s − 10.6·41-s + 1.30·43-s + 3.39·45-s + 13.2·47-s + 11.9·49-s + 15.9·51-s − 4.96·53-s + 5.59·55-s + ⋯
L(s)  = 1  − 1.23·3-s + 0.950·5-s − 1.64·7-s + 0.532·9-s + 0.792·11-s + 0.445·13-s − 1.17·15-s − 1.80·17-s + 0.229·19-s + 2.03·21-s + 1.32·23-s − 0.0962·25-s + 0.579·27-s − 1.26·29-s + 0.933·31-s − 0.981·33-s − 1.56·35-s + 1.52·37-s − 0.551·39-s − 1.65·41-s + 0.198·43-s + 0.505·45-s + 1.93·47-s + 1.71·49-s + 2.23·51-s − 0.681·53-s + 0.753·55-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 \( 1 + 2.14T + 3T^{2} \)
5 \( 1 - 2.12T + 5T^{2} \)
7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 - 1.60T + 13T^{2} \)
17 \( 1 + 7.43T + 17T^{2} \)
23 \( 1 - 6.34T + 23T^{2} \)
29 \( 1 + 6.78T + 29T^{2} \)
31 \( 1 - 5.19T + 31T^{2} \)
37 \( 1 - 9.25T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 1.30T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + 4.96T + 53T^{2} \)
59 \( 1 + 9.67T + 59T^{2} \)
61 \( 1 - 0.0771T + 61T^{2} \)
67 \( 1 - 8.62T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 6.36T + 73T^{2} \)
83 \( 1 + 0.269T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−7.34830905445518207897629658186, −6.66680158986309600458394048006, −6.21495867368614750393990451300, −5.92463561998311246756257841203, −4.97726206530320863403385024314, −4.16664879212762856896474869434, −3.20846265161551104120091311546, −2.30868972286550903036236798409, −1.08578522346039099856117117325, 0, 1.08578522346039099856117117325, 2.30868972286550903036236798409, 3.20846265161551104120091311546, 4.16664879212762856896474869434, 4.97726206530320863403385024314, 5.92463561998311246756257841203, 6.21495867368614750393990451300, 6.66680158986309600458394048006, 7.34830905445518207897629658186

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