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  − 1.94·3-s − 1.68·5-s − 1.54·7-s + 0.784·9-s − 2.23·11-s − 1.94·13-s + 3.28·15-s − 2.28·17-s + 19-s + 3.00·21-s + 0.628·23-s − 2.14·25-s + 4.30·27-s + 9.85·29-s + 8.81·31-s + 4.34·33-s + 2.60·35-s − 2.52·37-s + 3.78·39-s + 8.82·41-s − 2.39·43-s − 1.32·45-s − 1.22·47-s − 4.61·49-s + 4.44·51-s + 5.98·53-s + 3.77·55-s + ⋯
L(s)  = 1  − 1.12·3-s − 0.755·5-s − 0.583·7-s + 0.261·9-s − 0.673·11-s − 0.540·13-s + 0.848·15-s − 0.554·17-s + 0.229·19-s + 0.655·21-s + 0.131·23-s − 0.429·25-s + 0.829·27-s + 1.82·29-s + 1.58·31-s + 0.756·33-s + 0.441·35-s − 0.415·37-s + 0.606·39-s + 1.37·41-s − 0.365·43-s − 0.197·45-s − 0.178·47-s − 0.659·49-s + 0.622·51-s + 0.822·53-s + 0.508·55-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 \( 1 + 1.94T + 3T^{2} \)
5 \( 1 + 1.68T + 5T^{2} \)
7 \( 1 + 1.54T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 1.94T + 13T^{2} \)
17 \( 1 + 2.28T + 17T^{2} \)
23 \( 1 - 0.628T + 23T^{2} \)
29 \( 1 - 9.85T + 29T^{2} \)
31 \( 1 - 8.81T + 31T^{2} \)
37 \( 1 + 2.52T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + 1.22T + 47T^{2} \)
53 \( 1 - 5.98T + 53T^{2} \)
59 \( 1 + 8.90T + 59T^{2} \)
61 \( 1 + 0.227T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 0.0896T + 89T^{2} \)
97 \( 1 + 6.86T + 97T^{2} \)
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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

−7.75175867152268961584185978608, −6.77546109294162006474742763665, −6.41877133268097042976336298120, −5.57467421086692136079239036464, −4.82324989552928082245499709835, −4.31183031202181355836842097685, −3.17245560226099658308889935791, −2.48405525838619087878312605239, −0.877322514979659159441187585786, 0, 0.877322514979659159441187585786, 2.48405525838619087878312605239, 3.17245560226099658308889935791, 4.31183031202181355836842097685, 4.82324989552928082245499709835, 5.57467421086692136079239036464, 6.41877133268097042976336298120, 6.77546109294162006474742763665, 7.75175867152268961584185978608

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