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  + 2.79·3-s − 2.61·5-s + 0.0133·7-s + 4.82·9-s + 0.228·11-s + 0.688·13-s − 7.32·15-s − 1.16·17-s − 0.379·19-s + 0.0373·21-s + 7.37·23-s + 1.86·25-s + 5.09·27-s − 4.16·29-s + 5.39·31-s + 0.639·33-s − 0.0350·35-s + 8.04·37-s + 1.92·39-s + 4.81·41-s − 4.53·43-s − 12.6·45-s + 5.68·47-s − 6.99·49-s − 3.25·51-s + 7.16·53-s − 0.598·55-s + ⋯
L(s)  = 1  + 1.61·3-s − 1.17·5-s + 0.00505·7-s + 1.60·9-s + 0.0688·11-s + 0.190·13-s − 1.89·15-s − 0.282·17-s − 0.0871·19-s + 0.00815·21-s + 1.53·23-s + 0.372·25-s + 0.981·27-s − 0.772·29-s + 0.968·31-s + 0.111·33-s − 0.00591·35-s + 1.32·37-s + 0.308·39-s + 0.752·41-s − 0.691·43-s − 1.88·45-s + 0.828·47-s − 0.999·49-s − 0.456·51-s + 0.984·53-s − 0.0807·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.028541691$
$L(\frac12)$  $\approx$  $3.028541691$
$L(\frac{3}{2})$   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 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
7 \( 1 - 0.0133T + 7T^{2} \)
11 \( 1 - 0.228T + 11T^{2} \)
13 \( 1 - 0.688T + 13T^{2} \)
17 \( 1 + 1.16T + 17T^{2} \)
19 \( 1 + 0.379T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 5.39T + 31T^{2} \)
37 \( 1 - 8.04T + 37T^{2} \)
41 \( 1 - 4.81T + 41T^{2} \)
43 \( 1 + 4.53T + 43T^{2} \)
47 \( 1 - 5.68T + 47T^{2} \)
53 \( 1 - 7.16T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 3.14T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 2.77T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 - 6.29T + 83T^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 + 5.10T + 97T^{2} \)
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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

−8.073650844628417675096608482996, −7.57785902715300785852453914886, −7.02365970588039499458179463560, −6.10330817356846489420145839766, −4.85224875501287097628882568189, −4.23048113499422306356140636175, −3.51874073554025895220131101797, −2.94977562683412433478325547463, −2.08664306666273949705572739726, −0.853900460426784484379603272812, 0.853900460426784484379603272812, 2.08664306666273949705572739726, 2.94977562683412433478325547463, 3.51874073554025895220131101797, 4.23048113499422306356140636175, 4.85224875501287097628882568189, 6.10330817356846489420145839766, 7.02365970588039499458179463560, 7.57785902715300785852453914886, 8.073650844628417675096608482996

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