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  + 0.314·3-s − 4.24·5-s + 1.07·7-s − 2.90·9-s + 3.34·11-s − 1.31·13-s − 1.33·15-s − 2.75·17-s + 19-s + 0.338·21-s + 9.53·23-s + 13.0·25-s − 1.85·27-s − 8.28·29-s + 3.02·31-s + 1.05·33-s − 4.57·35-s + 4.48·37-s − 0.414·39-s − 2.36·41-s − 10.5·43-s + 12.3·45-s − 3.65·47-s − 5.83·49-s − 0.866·51-s + 11.8·53-s − 14.2·55-s + ⋯
L(s)  = 1  + 0.181·3-s − 1.89·5-s + 0.407·7-s − 0.967·9-s + 1.00·11-s − 0.365·13-s − 0.344·15-s − 0.668·17-s + 0.229·19-s + 0.0739·21-s + 1.98·23-s + 2.60·25-s − 0.357·27-s − 1.53·29-s + 0.542·31-s + 0.183·33-s − 0.773·35-s + 0.737·37-s − 0.0663·39-s − 0.369·41-s − 1.60·43-s + 1.83·45-s − 0.533·47-s − 0.834·49-s − 0.121·51-s + 1.62·53-s − 1.91·55-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 \( 1 - 0.314T + 3T^{2} \)
5 \( 1 + 4.24T + 5T^{2} \)
7 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
23 \( 1 - 9.53T + 23T^{2} \)
29 \( 1 + 8.28T + 29T^{2} \)
31 \( 1 - 3.02T + 31T^{2} \)
37 \( 1 - 4.48T + 37T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 3.65T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 8.61T + 61T^{2} \)
67 \( 1 - 3.47T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 3.96T + 73T^{2} \)
83 \( 1 - 7.24T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 0.407T + 97T^{2} \)
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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

−7.81225162712839840265451013724, −7.02857691766716646653058508608, −6.64850380253611370716341471174, −5.34218364352001922085090196439, −4.77971185396908089978861256440, −3.90511174851733833370294261718, −3.40863945416134198475346928804, −2.52937859402052583104151814497, −1.10929933429005821256921550193, 0, 1.10929933429005821256921550193, 2.52937859402052583104151814497, 3.40863945416134198475346928804, 3.90511174851733833370294261718, 4.77971185396908089978861256440, 5.34218364352001922085090196439, 6.64850380253611370716341471174, 7.02857691766716646653058508608, 7.81225162712839840265451013724

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