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  + 0.551·3-s − 2.33·5-s + 1.71·7-s − 2.69·9-s − 3.53·11-s + 4.69·13-s − 1.28·15-s + 5.32·17-s + 19-s + 0.945·21-s − 3.91·23-s + 0.440·25-s − 3.14·27-s − 0.324·29-s + 3.84·31-s − 1.95·33-s − 3.99·35-s − 2.10·37-s + 2.59·39-s − 11.2·41-s + 4.62·43-s + 6.28·45-s + 2.89·47-s − 4.06·49-s + 2.93·51-s − 0.546·53-s + 8.24·55-s + ⋯
L(s)  = 1  + 0.318·3-s − 1.04·5-s + 0.647·7-s − 0.898·9-s − 1.06·11-s + 1.30·13-s − 0.332·15-s + 1.29·17-s + 0.229·19-s + 0.206·21-s − 0.817·23-s + 0.0880·25-s − 0.604·27-s − 0.0603·29-s + 0.690·31-s − 0.339·33-s − 0.675·35-s − 0.345·37-s + 0.414·39-s − 1.76·41-s + 0.704·43-s + 0.937·45-s + 0.421·47-s − 0.580·49-s + 0.411·51-s − 0.0750·53-s + 1.11·55-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 \( 1 - 0.551T + 3T^{2} \)
5 \( 1 + 2.33T + 5T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
13 \( 1 - 4.69T + 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
23 \( 1 + 3.91T + 23T^{2} \)
29 \( 1 + 0.324T + 29T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + 2.10T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 + 0.546T + 53T^{2} \)
59 \( 1 - 0.300T + 59T^{2} \)
61 \( 1 - 9.09T + 61T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + 2.90T + 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
83 \( 1 + 7.18T + 83T^{2} \)
89 \( 1 + 0.382T + 89T^{2} \)
97 \( 1 - 9.48T + 97T^{2} \)
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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

−7.918932718787266530937825428864, −7.32326575596702770075972878192, −6.19375892566273243034437756398, −5.54483324590928608289375752518, −4.87405785494463579373072237789, −3.81107546624546022833385578266, −3.38207075722045803760135891281, −2.45596973089389271541625515594, −1.26960611677553576538793974657, 0, 1.26960611677553576538793974657, 2.45596973089389271541625515594, 3.38207075722045803760135891281, 3.81107546624546022833385578266, 4.87405785494463579373072237789, 5.54483324590928608289375752518, 6.19375892566273243034437756398, 7.32326575596702770075972878192, 7.918932718787266530937825428864

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