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  + 0.416·3-s + 3.95·5-s + 2.52·7-s − 2.82·9-s + 3.42·11-s − 3.31·13-s + 1.64·15-s − 7.23·17-s + 4.52·19-s + 1.05·21-s + 7.20·23-s + 10.6·25-s − 2.42·27-s + 7.29·29-s − 4.45·31-s + 1.42·33-s + 9.99·35-s − 6.53·37-s − 1.37·39-s + 4.03·41-s − 9.48·43-s − 11.1·45-s − 0.0683·47-s − 0.611·49-s − 3.01·51-s + 9.07·53-s + 13.5·55-s + ⋯
L(s)  = 1  + 0.240·3-s + 1.76·5-s + 0.955·7-s − 0.942·9-s + 1.03·11-s − 0.918·13-s + 0.425·15-s − 1.75·17-s + 1.03·19-s + 0.229·21-s + 1.50·23-s + 2.12·25-s − 0.466·27-s + 1.35·29-s − 0.799·31-s + 0.247·33-s + 1.69·35-s − 1.07·37-s − 0.220·39-s + 0.630·41-s − 1.44·43-s − 1.66·45-s − 0.00996·47-s − 0.0873·49-s − 0.421·51-s + 1.24·53-s + 1.82·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.496280327$
$L(\frac12)$  $\approx$  $3.496280327$
$L(\frac{3}{2})$   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 \( 1 - 0.416T + 3T^{2} \)
5 \( 1 - 3.95T + 5T^{2} \)
7 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 - 3.42T + 11T^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
17 \( 1 + 7.23T + 17T^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 - 7.20T + 23T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
37 \( 1 + 6.53T + 37T^{2} \)
41 \( 1 - 4.03T + 41T^{2} \)
43 \( 1 + 9.48T + 43T^{2} \)
47 \( 1 + 0.0683T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 - 8.40T + 59T^{2} \)
61 \( 1 - 3.48T + 61T^{2} \)
67 \( 1 - 7.25T + 67T^{2} \)
71 \( 1 - 3.26T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + 8.80T + 89T^{2} \)
97 \( 1 + 2.44T + 97T^{2} \)
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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

−8.444860810476527008860027130776, −7.06728598492547995668704777192, −6.77674756967391729092558435629, −5.90046490761387477274770920225, −5.02636947347515551687063762906, −4.90251743535360227197705789011, −3.49133436346005585440368642805, −2.48037811177264595325082696777, −2.03070564627872276140439879213, −1.01711639801331800636952070117, 1.01711639801331800636952070117, 2.03070564627872276140439879213, 2.48037811177264595325082696777, 3.49133436346005585440368642805, 4.90251743535360227197705789011, 5.02636947347515551687063762906, 5.90046490761387477274770920225, 6.77674756967391729092558435629, 7.06728598492547995668704777192, 8.444860810476527008860027130776

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