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  + 1.63·3-s − 0.519·5-s − 1.27·7-s − 0.337·9-s + 0.383·11-s − 3.76·13-s − 0.848·15-s − 2.19·17-s + 8.06·19-s − 2.08·21-s + 8.44·23-s − 4.72·25-s − 5.44·27-s + 1.75·29-s + 2.47·31-s + 0.624·33-s + 0.664·35-s − 5.22·37-s − 6.13·39-s − 8.06·41-s − 2.28·43-s + 0.175·45-s + 6.12·47-s − 5.36·49-s − 3.58·51-s − 7.72·53-s − 0.199·55-s + ⋯
L(s)  = 1  + 0.942·3-s − 0.232·5-s − 0.483·7-s − 0.112·9-s + 0.115·11-s − 1.04·13-s − 0.219·15-s − 0.532·17-s + 1.85·19-s − 0.455·21-s + 1.76·23-s − 0.945·25-s − 1.04·27-s + 0.325·29-s + 0.444·31-s + 0.108·33-s + 0.112·35-s − 0.858·37-s − 0.982·39-s − 1.26·41-s − 0.348·43-s + 0.0261·45-s + 0.892·47-s − 0.766·49-s − 0.501·51-s − 1.06·53-s − 0.0268·55-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 \( 1 - 1.63T + 3T^{2} \)
5 \( 1 + 0.519T + 5T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
11 \( 1 - 0.383T + 11T^{2} \)
13 \( 1 + 3.76T + 13T^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 - 8.06T + 19T^{2} \)
23 \( 1 - 8.44T + 23T^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 + 8.06T + 41T^{2} \)
43 \( 1 + 2.28T + 43T^{2} \)
47 \( 1 - 6.12T + 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 3.82T + 71T^{2} \)
73 \( 1 - 4.88T + 73T^{2} \)
79 \( 1 + 4.92T + 79T^{2} \)
83 \( 1 - 6.10T + 83T^{2} \)
89 \( 1 + 0.946T + 89T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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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

−7.77498113926330988145807129264, −7.13093006220965791431948447291, −6.52910628994797438436048910324, −5.38746666519678671714234357362, −4.91388181172447075819103100738, −3.79576530094697319289606743622, −3.10255959334839360546861013928, −2.62201206952368876414025285717, −1.44165691525939651156520100861, 0, 1.44165691525939651156520100861, 2.62201206952368876414025285717, 3.10255959334839360546861013928, 3.79576530094697319289606743622, 4.91388181172447075819103100738, 5.38746666519678671714234357362, 6.52910628994797438436048910324, 7.13093006220965791431948447291, 7.77498113926330988145807129264

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