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  − 0.217·3-s + 1.76·5-s + 1.63·7-s − 2.95·9-s − 4.73·11-s + 4.29·13-s − 0.383·15-s − 6.17·17-s + 3.23·19-s − 0.354·21-s − 5.20·23-s − 1.89·25-s + 1.29·27-s − 5.37·29-s + 3.65·31-s + 1.02·33-s + 2.87·35-s + 10.2·37-s − 0.933·39-s + 12.3·41-s − 4.50·43-s − 5.20·45-s + 11.0·47-s − 4.33·49-s + 1.34·51-s − 2.84·53-s − 8.34·55-s + ⋯
L(s)  = 1  − 0.125·3-s + 0.788·5-s + 0.617·7-s − 0.984·9-s − 1.42·11-s + 1.19·13-s − 0.0989·15-s − 1.49·17-s + 0.742·19-s − 0.0774·21-s − 1.08·23-s − 0.378·25-s + 0.248·27-s − 0.997·29-s + 0.655·31-s + 0.179·33-s + 0.486·35-s + 1.68·37-s − 0.149·39-s + 1.92·41-s − 0.686·43-s − 0.776·45-s + 1.61·47-s − 0.619·49-s + 0.187·51-s − 0.390·53-s − 1.12·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.855923265$
$L(\frac12)$  $\approx$  $1.855923265$
$L(\frac{3}{2})$   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 \( 1 + 0.217T + 3T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 - 3.23T + 19T^{2} \)
23 \( 1 + 5.20T + 23T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 4.50T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + 0.223T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 7.71T + 67T^{2} \)
71 \( 1 + 5.66T + 71T^{2} \)
73 \( 1 - 6.65T + 73T^{2} \)
79 \( 1 - 8.73T + 79T^{2} \)
83 \( 1 + 1.13T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.993230307066145936300773873579, −7.64005049895708087594073084276, −6.34936386848139256726197404004, −5.95084866905485137212321671341, −5.36098646488719307012978057465, −4.56666040021401909985474544102, −3.64061960636620159590588200598, −2.48659532301624752072255583420, −2.10942882749527395757841154690, −0.69853009255750439657647727298, 0.69853009255750439657647727298, 2.10942882749527395757841154690, 2.48659532301624752072255583420, 3.64061960636620159590588200598, 4.56666040021401909985474544102, 5.36098646488719307012978057465, 5.95084866905485137212321671341, 6.34936386848139256726197404004, 7.64005049895708087594073084276, 7.993230307066145936300773873579

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