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  − 2.56·3-s − 2.44·5-s − 1.80·7-s + 3.58·9-s − 2.13·11-s + 4.47·13-s + 6.28·15-s − 0.551·17-s + 1.19·19-s + 4.64·21-s + 0.789·23-s + 0.994·25-s − 1.50·27-s − 4.78·29-s − 4.84·31-s + 5.48·33-s + 4.42·35-s − 0.0175·37-s − 11.4·39-s + 6.24·41-s − 2.48·43-s − 8.78·45-s − 13.2·47-s − 3.72·49-s + 1.41·51-s + 4.32·53-s + 5.23·55-s + ⋯
L(s)  = 1  − 1.48·3-s − 1.09·5-s − 0.683·7-s + 1.19·9-s − 0.644·11-s + 1.24·13-s + 1.62·15-s − 0.133·17-s + 0.274·19-s + 1.01·21-s + 0.164·23-s + 0.198·25-s − 0.290·27-s − 0.889·29-s − 0.869·31-s + 0.954·33-s + 0.748·35-s − 0.00288·37-s − 1.83·39-s + 0.974·41-s − 0.378·43-s − 1.30·45-s − 1.93·47-s − 0.532·49-s + 0.198·51-s + 0.593·53-s + 0.705·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3063594482$
$L(\frac12)$  $\approx$  $0.3063594482$
$L(\frac{3}{2})$   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 \( 1 + 2.56T + 3T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 1.80T + 7T^{2} \)
11 \( 1 + 2.13T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 0.551T + 17T^{2} \)
19 \( 1 - 1.19T + 19T^{2} \)
23 \( 1 - 0.789T + 23T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 + 0.0175T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 - 4.32T + 53T^{2} \)
59 \( 1 + 1.86T + 59T^{2} \)
61 \( 1 - 1.93T + 61T^{2} \)
67 \( 1 + 5.88T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 0.170T + 89T^{2} \)
97 \( 1 + 2.90T + 97T^{2} \)
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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

−7.922925156575258959313037049043, −7.27458928439766135898793430167, −6.56730231736573446700234095721, −5.92273805382148445242483373970, −5.35068716309594169018353984755, −4.48824465014648544953460693520, −3.75972613898701892898222332098, −3.03824280367237469311098980836, −1.50030501463659238638669718340, −0.32450503562078242513419484490, 0.32450503562078242513419484490, 1.50030501463659238638669718340, 3.03824280367237469311098980836, 3.75972613898701892898222332098, 4.48824465014648544953460693520, 5.35068716309594169018353984755, 5.92273805382148445242483373970, 6.56730231736573446700234095721, 7.27458928439766135898793430167, 7.922925156575258959313037049043

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