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  − 2.32·3-s + 3.89·5-s + 0.0996·7-s + 2.39·9-s − 5.90·11-s − 0.114·13-s − 9.04·15-s − 0.598·17-s − 3.75·19-s − 0.231·21-s + 8.45·23-s + 10.1·25-s + 1.40·27-s − 4.82·29-s − 5.02·31-s + 13.7·33-s + 0.388·35-s + 4.30·37-s + 0.265·39-s − 1.72·41-s + 7.69·43-s + 9.33·45-s + 3.57·47-s − 6.99·49-s + 1.39·51-s + 8.09·53-s − 23.0·55-s + ⋯
L(s)  = 1  − 1.34·3-s + 1.74·5-s + 0.0376·7-s + 0.798·9-s − 1.78·11-s − 0.0316·13-s − 2.33·15-s − 0.145·17-s − 0.862·19-s − 0.0505·21-s + 1.76·23-s + 2.03·25-s + 0.270·27-s − 0.896·29-s − 0.901·31-s + 2.38·33-s + 0.0656·35-s + 0.708·37-s + 0.0424·39-s − 0.270·41-s + 1.17·43-s + 1.39·45-s + 0.521·47-s − 0.998·49-s + 0.194·51-s + 1.11·53-s − 3.10·55-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 \( 1 + 2.32T + 3T^{2} \)
5 \( 1 - 3.89T + 5T^{2} \)
7 \( 1 - 0.0996T + 7T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
13 \( 1 + 0.114T + 13T^{2} \)
17 \( 1 + 0.598T + 17T^{2} \)
19 \( 1 + 3.75T + 19T^{2} \)
23 \( 1 - 8.45T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 5.02T + 31T^{2} \)
37 \( 1 - 4.30T + 37T^{2} \)
41 \( 1 + 1.72T + 41T^{2} \)
43 \( 1 - 7.69T + 43T^{2} \)
47 \( 1 - 3.57T + 47T^{2} \)
53 \( 1 - 8.09T + 53T^{2} \)
59 \( 1 + 8.16T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 0.648T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 + 3.11T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 4.53T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.43186950972058296707814469175, −6.89565075539963675510064971119, −5.98398774382589369938863284693, −5.64440925932813082283833781174, −5.15419406183325686667051281596, −4.46835448455907992664960744690, −2.91536004726570315944503871953, −2.29227200539071883718182257369, −1.24786530704091697553979774520, 0, 1.24786530704091697553979774520, 2.29227200539071883718182257369, 2.91536004726570315944503871953, 4.46835448455907992664960744690, 5.15419406183325686667051281596, 5.64440925932813082283833781174, 5.98398774382589369938863284693, 6.89565075539963675510064971119, 7.43186950972058296707814469175

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