Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.22·3-s + 2.92·5-s + 0.525·7-s + 7.41·9-s − 1.19·11-s − 1.45·13-s − 9.44·15-s + 4.32·17-s + 2.66·19-s − 1.69·21-s − 0.287·23-s + 3.56·25-s − 14.2·27-s + 4.93·29-s − 5.71·31-s + 3.85·33-s + 1.53·35-s − 3.78·37-s + 4.70·39-s + 10.2·41-s + 3.99·43-s + 21.7·45-s + 1.23·47-s − 6.72·49-s − 13.9·51-s + 12.0·53-s − 3.49·55-s + ⋯
L(s)  = 1  − 1.86·3-s + 1.30·5-s + 0.198·7-s + 2.47·9-s − 0.359·11-s − 0.404·13-s − 2.43·15-s + 1.04·17-s + 0.612·19-s − 0.370·21-s − 0.0599·23-s + 0.713·25-s − 2.74·27-s + 0.916·29-s − 1.02·31-s + 0.670·33-s + 0.260·35-s − 0.622·37-s + 0.753·39-s + 1.60·41-s + 0.609·43-s + 3.23·45-s + 0.180·47-s − 0.960·49-s − 1.95·51-s + 1.65·53-s − 0.471·55-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  =  0
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.416052945$
$L(\frac12)$  $\approx$  $1.416052945$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 3.22T + 3T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
7 \( 1 - 0.525T + 7T^{2} \)
11 \( 1 + 1.19T + 11T^{2} \)
13 \( 1 + 1.45T + 13T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 2.66T + 19T^{2} \)
23 \( 1 + 0.287T + 23T^{2} \)
29 \( 1 - 4.93T + 29T^{2} \)
31 \( 1 + 5.71T + 31T^{2} \)
37 \( 1 + 3.78T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 3.99T + 43T^{2} \)
47 \( 1 - 1.23T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 - 2.87T + 67T^{2} \)
71 \( 1 + 4.40T + 71T^{2} \)
73 \( 1 - 1.18T + 73T^{2} \)
79 \( 1 + 8.82T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 7.97T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.77952981047197247303264636781, −7.18393225015834319382112444620, −6.46062726930784189729706661779, −5.71172576299614946436324248439, −5.46172838480477259768089615922, −4.85257146185517453402788261016, −3.88636629300685484810617248046, −2.56317943017636244032168417314, −1.56066825157600964318028581258, −0.73601396513161070328124010235, 0.73601396513161070328124010235, 1.56066825157600964318028581258, 2.56317943017636244032168417314, 3.88636629300685484810617248046, 4.85257146185517453402788261016, 5.46172838480477259768089615922, 5.71172576299614946436324248439, 6.46062726930784189729706661779, 7.18393225015834319382112444620, 7.77952981047197247303264636781

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