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  + 0.823·3-s − 1.53·5-s + 0.257·7-s − 2.32·9-s + 1.07·11-s − 6.44·13-s − 1.26·15-s + 3.61·17-s − 2.46·19-s + 0.212·21-s + 6.43·23-s − 2.64·25-s − 4.38·27-s + 6.41·29-s − 4.45·31-s + 0.884·33-s − 0.395·35-s − 5.37·37-s − 5.30·39-s − 1.12·41-s + 5.77·43-s + 3.56·45-s + 10.2·47-s − 6.93·49-s + 2.97·51-s − 3.86·53-s − 1.64·55-s + ⋯
L(s)  = 1  + 0.475·3-s − 0.685·5-s + 0.0975·7-s − 0.773·9-s + 0.323·11-s − 1.78·13-s − 0.326·15-s + 0.875·17-s − 0.565·19-s + 0.0463·21-s + 1.34·23-s − 0.529·25-s − 0.843·27-s + 1.19·29-s − 0.799·31-s + 0.153·33-s − 0.0668·35-s − 0.883·37-s − 0.849·39-s − 0.176·41-s + 0.880·43-s + 0.530·45-s + 1.49·47-s − 0.990·49-s + 0.416·51-s − 0.530·53-s − 0.222·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.435419175$
$L(\frac12)$  $\approx$  $1.435419175$
$L(\frac{3}{2})$   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 \( 1 - 0.823T + 3T^{2} \)
5 \( 1 + 1.53T + 5T^{2} \)
7 \( 1 - 0.257T + 7T^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + 6.44T + 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 - 6.43T + 23T^{2} \)
29 \( 1 - 6.41T + 29T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
37 \( 1 + 5.37T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 5.77T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 3.86T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 5.43T + 61T^{2} \)
67 \( 1 + 3.75T + 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 + 9.39T + 73T^{2} \)
79 \( 1 + 3.75T + 79T^{2} \)
83 \( 1 - 7.37T + 83T^{2} \)
89 \( 1 + 2.01T + 89T^{2} \)
97 \( 1 - 0.889T + 97T^{2} \)
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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

−8.070481609578446577801398323350, −7.39285684159969416339188322429, −6.95186413099779849848811336062, −5.85564020576909768474244835140, −5.14666050320657701735822605079, −4.43606505833640439078488434395, −3.52039559049303967985960381396, −2.84983571730022023561229195006, −2.04309350738226895027188660685, −0.59007097301929066974401161657, 0.59007097301929066974401161657, 2.04309350738226895027188660685, 2.84983571730022023561229195006, 3.52039559049303967985960381396, 4.43606505833640439078488434395, 5.14666050320657701735822605079, 5.85564020576909768474244835140, 6.95186413099779849848811336062, 7.39285684159969416339188322429, 8.070481609578446577801398323350

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