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  − 0.345·3-s + 2.40·5-s − 1.76·7-s − 2.88·9-s − 2.38·11-s + 4.86·13-s − 0.832·15-s + 4.36·17-s − 4.69·19-s + 0.610·21-s − 1.71·23-s + 0.799·25-s + 2.03·27-s + 2.59·29-s − 0.727·31-s + 0.824·33-s − 4.25·35-s − 7.06·37-s − 1.68·39-s − 0.505·41-s + 5.98·43-s − 6.93·45-s − 8.49·47-s − 3.88·49-s − 1.50·51-s + 4.93·53-s − 5.74·55-s + ⋯
L(s)  = 1  − 0.199·3-s + 1.07·5-s − 0.667·7-s − 0.960·9-s − 0.718·11-s + 1.34·13-s − 0.214·15-s + 1.05·17-s − 1.07·19-s + 0.133·21-s − 0.356·23-s + 0.159·25-s + 0.391·27-s + 0.482·29-s − 0.130·31-s + 0.143·33-s − 0.718·35-s − 1.16·37-s − 0.269·39-s − 0.0789·41-s + 0.912·43-s − 1.03·45-s − 1.23·47-s − 0.554·49-s − 0.211·51-s + 0.678·53-s − 0.774·55-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 \( 1 + 0.345T + 3T^{2} \)
5 \( 1 - 2.40T + 5T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 + 2.38T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 + 4.69T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 - 2.59T + 29T^{2} \)
31 \( 1 + 0.727T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + 0.505T + 41T^{2} \)
43 \( 1 - 5.98T + 43T^{2} \)
47 \( 1 + 8.49T + 47T^{2} \)
53 \( 1 - 4.93T + 53T^{2} \)
59 \( 1 + 7.80T + 59T^{2} \)
61 \( 1 - 6.09T + 61T^{2} \)
67 \( 1 + 0.910T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 7.63T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−7.88841624648194322354784755661, −6.72000746125402245559554326851, −6.21303404673421379405357049404, −5.68173592751908298717170875008, −5.14092729791772968078439515758, −3.91668490639326121875542716921, −3.14551955983788669850201388221, −2.37640658032411415920476645294, −1.36302550940283962618178374990, 0, 1.36302550940283962618178374990, 2.37640658032411415920476645294, 3.14551955983788669850201388221, 3.91668490639326121875542716921, 5.14092729791772968078439515758, 5.68173592751908298717170875008, 6.21303404673421379405357049404, 6.72000746125402245559554326851, 7.88841624648194322354784755661

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