Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.80·3-s + 0.563·5-s − 3.54·7-s + 4.87·9-s + 0.185·11-s + 1.50·13-s + 1.58·15-s − 4.26·17-s − 5.87·19-s − 9.94·21-s − 4.13·23-s − 4.68·25-s + 5.25·27-s + 6.53·29-s − 9.18·31-s + 0.519·33-s − 1.99·35-s + 2.36·37-s + 4.23·39-s − 1.06·41-s + 9.99·43-s + 2.74·45-s + 6.05·47-s + 5.56·49-s − 11.9·51-s − 6.60·53-s + 0.104·55-s + ⋯
L(s)  = 1  + 1.61·3-s + 0.251·5-s − 1.34·7-s + 1.62·9-s + 0.0558·11-s + 0.418·13-s + 0.408·15-s − 1.03·17-s − 1.34·19-s − 2.17·21-s − 0.862·23-s − 0.936·25-s + 1.01·27-s + 1.21·29-s − 1.65·31-s + 0.0904·33-s − 0.337·35-s + 0.389·37-s + 0.677·39-s − 0.166·41-s + 1.52·43-s + 0.409·45-s + 0.883·47-s + 0.795·49-s − 1.67·51-s − 0.907·53-s + 0.0140·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;751\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 2.80T + 3T^{2} \)
5 \( 1 - 0.563T + 5T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 - 0.185T + 11T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 + 4.26T + 17T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 + 4.13T + 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 + 9.18T + 31T^{2} \)
37 \( 1 - 2.36T + 37T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 - 9.99T + 43T^{2} \)
47 \( 1 - 6.05T + 47T^{2} \)
53 \( 1 + 6.60T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 6.98T + 61T^{2} \)
67 \( 1 + 8.95T + 67T^{2} \)
71 \( 1 + 5.32T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 5.38T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 5.71T + 97T^{2} \)
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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

−7.79524960414527317252735186801, −7.14764148194007388423404939761, −6.31427336187892105191909462226, −5.89521832990292879651398838860, −4.35338730566431136892912739339, −3.98116728875870514251600090013, −3.11451778025107588054251625336, −2.45033744260081653039387558470, −1.73129865917974765233080451338, 0, 1.73129865917974765233080451338, 2.45033744260081653039387558470, 3.11451778025107588054251625336, 3.98116728875870514251600090013, 4.35338730566431136892912739339, 5.89521832990292879651398838860, 6.31427336187892105191909462226, 7.14764148194007388423404939761, 7.79524960414527317252735186801

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