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  + 0.363·3-s + 1.56·5-s + 3.79·7-s − 2.86·9-s − 0.247·11-s + 4.99·13-s + 0.568·15-s + 5.01·17-s − 8.47·19-s + 1.37·21-s + 5.06·23-s − 2.54·25-s − 2.13·27-s + 0.226·29-s − 3.67·31-s − 0.0900·33-s + 5.93·35-s + 2.82·37-s + 1.81·39-s + 4.03·41-s + 3.40·43-s − 4.49·45-s + 7.95·47-s + 7.36·49-s + 1.82·51-s + 4.00·53-s − 0.388·55-s + ⋯
L(s)  = 1  + 0.209·3-s + 0.700·5-s + 1.43·7-s − 0.956·9-s − 0.0747·11-s + 1.38·13-s + 0.146·15-s + 1.21·17-s − 1.94·19-s + 0.300·21-s + 1.05·23-s − 0.509·25-s − 0.410·27-s + 0.0421·29-s − 0.660·31-s − 0.0156·33-s + 1.00·35-s + 0.463·37-s + 0.290·39-s + 0.629·41-s + 0.519·43-s − 0.669·45-s + 1.16·47-s + 1.05·49-s + 0.254·51-s + 0.549·53-s − 0.0523·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.073414314$
$L(\frac12)$  $\approx$  $3.073414314$
$L(\frac{3}{2})$   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 \( 1 - 0.363T + 3T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
7 \( 1 - 3.79T + 7T^{2} \)
11 \( 1 + 0.247T + 11T^{2} \)
13 \( 1 - 4.99T + 13T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + 8.47T + 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 - 0.226T + 29T^{2} \)
31 \( 1 + 3.67T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 - 4.03T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 - 7.95T + 47T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
59 \( 1 - 8.61T + 59T^{2} \)
61 \( 1 + 1.44T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 0.864T + 71T^{2} \)
73 \( 1 + 2.13T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 1.16T + 97T^{2} \)
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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

−8.237651388070927168082799017339, −7.55384332065175010703449462325, −6.55563620806230843551986417264, −5.67696923524560285730454918673, −5.52943576618902447553142739654, −4.40522753887020882894791323216, −3.70387683705764322887959401563, −2.60662565549848915420835157957, −1.88844098368698458441608112200, −0.963489112988567521923774120561, 0.963489112988567521923774120561, 1.88844098368698458441608112200, 2.60662565549848915420835157957, 3.70387683705764322887959401563, 4.40522753887020882894791323216, 5.52943576618902447553142739654, 5.67696923524560285730454918673, 6.55563620806230843551986417264, 7.55384332065175010703449462325, 8.237651388070927168082799017339

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