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  + 1.38·3-s − 3.43·5-s + 3.30·7-s − 1.09·9-s − 0.715·11-s + 5.15·13-s − 4.73·15-s − 1.52·17-s + 7.44·19-s + 4.56·21-s − 3.93·23-s + 6.78·25-s − 5.65·27-s − 4.20·29-s + 6.78·31-s − 0.987·33-s − 11.3·35-s − 5.07·37-s + 7.11·39-s + 1.57·41-s − 1.14·43-s + 3.75·45-s + 4.06·47-s + 3.93·49-s − 2.10·51-s + 3.64·53-s + 2.45·55-s + ⋯
L(s)  = 1  + 0.796·3-s − 1.53·5-s + 1.24·7-s − 0.364·9-s − 0.215·11-s + 1.42·13-s − 1.22·15-s − 0.370·17-s + 1.70·19-s + 0.996·21-s − 0.821·23-s + 1.35·25-s − 1.08·27-s − 0.780·29-s + 1.21·31-s − 0.171·33-s − 1.91·35-s − 0.835·37-s + 1.13·39-s + 0.245·41-s − 0.174·43-s + 0.560·45-s + 0.593·47-s + 0.562·49-s − 0.295·51-s + 0.500·53-s + 0.331·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.261951872$
$L(\frac12)$  $\approx$  $2.261951872$
$L(\frac{3}{2})$   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 \( 1 - 1.38T + 3T^{2} \)
5 \( 1 + 3.43T + 5T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + 0.715T + 11T^{2} \)
13 \( 1 - 5.15T + 13T^{2} \)
17 \( 1 + 1.52T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + 3.93T + 23T^{2} \)
29 \( 1 + 4.20T + 29T^{2} \)
31 \( 1 - 6.78T + 31T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 + 1.14T + 43T^{2} \)
47 \( 1 - 4.06T + 47T^{2} \)
53 \( 1 - 3.64T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 5.54T + 61T^{2} \)
67 \( 1 + 2.31T + 67T^{2} \)
71 \( 1 - 6.03T + 71T^{2} \)
73 \( 1 + 6.14T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 8.06T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 - 4.13T + 97T^{2} \)
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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

−8.000755216097491021909777584786, −7.76116608032384691395612733673, −6.95568309599235798775109093024, −5.83517456939965704922926994396, −5.11815728588047127479625725280, −4.19709604131241306281598785938, −3.66884404641986038490317098069, −2.98712294507398582787980550506, −1.86864412156190462960282737316, −0.77855816035910049001272546494, 0.77855816035910049001272546494, 1.86864412156190462960282737316, 2.98712294507398582787980550506, 3.66884404641986038490317098069, 4.19709604131241306281598785938, 5.11815728588047127479625725280, 5.83517456939965704922926994396, 6.95568309599235798775109093024, 7.76116608032384691395612733673, 8.000755216097491021909777584786

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