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  − 2.24·3-s + 3.86·5-s − 4.12·7-s + 2.04·9-s − 3.58·11-s − 2.00·13-s − 8.68·15-s − 1.02·17-s + 2.57·19-s + 9.26·21-s − 6.19·23-s + 9.96·25-s + 2.14·27-s − 4.21·29-s − 3.01·31-s + 8.05·33-s − 15.9·35-s − 3.89·37-s + 4.51·39-s − 5.80·41-s + 0.0570·43-s + 7.90·45-s + 1.46·47-s + 10.0·49-s + 2.29·51-s + 1.31·53-s − 13.8·55-s + ⋯
L(s)  = 1  − 1.29·3-s + 1.72·5-s − 1.55·7-s + 0.680·9-s − 1.08·11-s − 0.557·13-s − 2.24·15-s − 0.247·17-s + 0.590·19-s + 2.02·21-s − 1.29·23-s + 1.99·25-s + 0.413·27-s − 0.782·29-s − 0.541·31-s + 1.40·33-s − 2.69·35-s − 0.640·37-s + 0.722·39-s − 0.906·41-s + 0.00869·43-s + 1.17·45-s + 0.213·47-s + 1.43·49-s + 0.321·51-s + 0.180·53-s − 1.87·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6994992106$
$L(\frac12)$  $\approx$  $0.6994992106$
$L(\frac{3}{2})$   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 \( 1 + 2.24T + 3T^{2} \)
5 \( 1 - 3.86T + 5T^{2} \)
7 \( 1 + 4.12T + 7T^{2} \)
11 \( 1 + 3.58T + 11T^{2} \)
13 \( 1 + 2.00T + 13T^{2} \)
17 \( 1 + 1.02T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 + 4.21T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
37 \( 1 + 3.89T + 37T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 - 0.0570T + 43T^{2} \)
47 \( 1 - 1.46T + 47T^{2} \)
53 \( 1 - 1.31T + 53T^{2} \)
59 \( 1 + 7.55T + 59T^{2} \)
61 \( 1 + 0.242T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 3.56T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−7.986719447212244406174617817769, −6.91281078024548884954563957655, −6.57761931536787207781488211946, −5.75775184731609717382651621001, −5.54173750828329017032234335462, −4.86450613461550475806046978133, −3.55855279455309288306053597846, −2.63964006591211346783181580721, −1.88372919856584405046930571087, −0.44373293668604843728629885927, 0.44373293668604843728629885927, 1.88372919856584405046930571087, 2.63964006591211346783181580721, 3.55855279455309288306053597846, 4.86450613461550475806046978133, 5.54173750828329017032234335462, 5.75775184731609717382651621001, 6.57761931536787207781488211946, 6.91281078024548884954563957655, 7.986719447212244406174617817769

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