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  + 2.46·3-s − 2.98·5-s + 1.48·7-s + 3.06·9-s + 3.74·11-s − 5.67·13-s − 7.34·15-s + 0.776·17-s + 19-s + 3.65·21-s − 4.98·23-s + 3.89·25-s + 0.169·27-s − 1.20·29-s − 6.15·31-s + 9.22·33-s − 4.42·35-s − 4.57·37-s − 13.9·39-s − 6.38·41-s + 7.04·43-s − 9.15·45-s + 4.91·47-s − 4.79·49-s + 1.91·51-s − 12.3·53-s − 11.1·55-s + ⋯
L(s)  = 1  + 1.42·3-s − 1.33·5-s + 0.561·7-s + 1.02·9-s + 1.12·11-s − 1.57·13-s − 1.89·15-s + 0.188·17-s + 0.229·19-s + 0.798·21-s − 1.03·23-s + 0.779·25-s + 0.0326·27-s − 0.223·29-s − 1.10·31-s + 1.60·33-s − 0.748·35-s − 0.751·37-s − 2.23·39-s − 0.997·41-s + 1.07·43-s − 1.36·45-s + 0.716·47-s − 0.685·49-s + 0.267·51-s − 1.70·53-s − 1.50·55-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 \( 1 - 2.46T + 3T^{2} \)
5 \( 1 + 2.98T + 5T^{2} \)
7 \( 1 - 1.48T + 7T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 - 0.776T + 17T^{2} \)
23 \( 1 + 4.98T + 23T^{2} \)
29 \( 1 + 1.20T + 29T^{2} \)
31 \( 1 + 6.15T + 31T^{2} \)
37 \( 1 + 4.57T + 37T^{2} \)
41 \( 1 + 6.38T + 41T^{2} \)
43 \( 1 - 7.04T + 43T^{2} \)
47 \( 1 - 4.91T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 8.82T + 61T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 - 0.725T + 71T^{2} \)
73 \( 1 - 5.70T + 73T^{2} \)
83 \( 1 + 6.75T + 83T^{2} \)
89 \( 1 + 2.60T + 89T^{2} \)
97 \( 1 - 0.185T + 97T^{2} \)
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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

−7.68913227766719283650523797656, −7.44789115515153328988926512767, −6.63292140026510009820148947457, −5.39189138624248857595698690876, −4.52269710750702422597665690752, −3.87931632452884098345788196061, −3.37896060210336440881452473686, −2.39487454252919006358560006364, −1.57482760596646002229890479813, 0, 1.57482760596646002229890479813, 2.39487454252919006358560006364, 3.37896060210336440881452473686, 3.87931632452884098345788196061, 4.52269710750702422597665690752, 5.39189138624248857595698690876, 6.63292140026510009820148947457, 7.44789115515153328988926512767, 7.68913227766719283650523797656

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