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  − 0.604·3-s + 2.44·5-s + 2.35·7-s − 2.63·9-s + 0.468·11-s + 3.50·13-s − 1.48·15-s − 6.18·17-s + 19-s − 1.42·21-s − 7.62·23-s + 1.00·25-s + 3.40·27-s + 1.30·29-s − 1.64·31-s − 0.283·33-s + 5.76·35-s − 9.36·37-s − 2.12·39-s − 0.982·41-s − 10.0·43-s − 6.45·45-s − 2.43·47-s − 1.45·49-s + 3.73·51-s + 2.13·53-s + 1.14·55-s + ⋯
L(s)  = 1  − 0.349·3-s + 1.09·5-s + 0.889·7-s − 0.877·9-s + 0.141·11-s + 0.973·13-s − 0.382·15-s − 1.49·17-s + 0.229·19-s − 0.310·21-s − 1.58·23-s + 0.200·25-s + 0.655·27-s + 0.241·29-s − 0.295·31-s − 0.0493·33-s + 0.974·35-s − 1.53·37-s − 0.339·39-s − 0.153·41-s − 1.52·43-s − 0.961·45-s − 0.355·47-s − 0.208·49-s + 0.523·51-s + 0.293·53-s + 0.154·55-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 \( 1 + 0.604T + 3T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 - 2.35T + 7T^{2} \)
11 \( 1 - 0.468T + 11T^{2} \)
13 \( 1 - 3.50T + 13T^{2} \)
17 \( 1 + 6.18T + 17T^{2} \)
23 \( 1 + 7.62T + 23T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + 9.36T + 37T^{2} \)
41 \( 1 + 0.982T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 - 2.13T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 6.19T + 61T^{2} \)
67 \( 1 - 9.26T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
83 \( 1 + 2.00T + 83T^{2} \)
89 \( 1 - 3.76T + 89T^{2} \)
97 \( 1 + 4.45T + 97T^{2} \)
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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

−7.929727497572738355783550817825, −6.69422262954993921176163340925, −6.31120999980599225800133167917, −5.59010124536290363251790195586, −5.02102600306288352503478526535, −4.15869389139994752861573080138, −3.16751524979417497651927889043, −2.04366333495759929387551151368, −1.59195134373130681617193127889, 0, 1.59195134373130681617193127889, 2.04366333495759929387551151368, 3.16751524979417497651927889043, 4.15869389139994752861573080138, 5.02102600306288352503478526535, 5.59010124536290363251790195586, 6.31120999980599225800133167917, 6.69422262954993921176163340925, 7.929727497572738355783550817825

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