Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.63·3-s + 2.82·5-s + 2.24·7-s − 0.321·9-s − 3.70·11-s − 6.16·13-s + 4.62·15-s − 5.15·17-s − 0.846·19-s + 3.67·21-s − 4.10·23-s + 2.97·25-s − 5.43·27-s − 1.81·29-s − 4.67·31-s − 6.05·33-s + 6.34·35-s − 2.39·37-s − 10.0·39-s + 2.12·41-s + 3.93·43-s − 0.907·45-s + 10.7·47-s − 1.94·49-s − 8.44·51-s + 6.85·53-s − 10.4·55-s + ⋯
L(s)  = 1  + 0.944·3-s + 1.26·5-s + 0.849·7-s − 0.107·9-s − 1.11·11-s − 1.70·13-s + 1.19·15-s − 1.25·17-s − 0.194·19-s + 0.802·21-s − 0.856·23-s + 0.594·25-s − 1.04·27-s − 0.336·29-s − 0.839·31-s − 1.05·33-s + 1.07·35-s − 0.394·37-s − 1.61·39-s + 0.331·41-s + 0.599·43-s − 0.135·45-s + 1.56·47-s − 0.278·49-s − 1.18·51-s + 0.942·53-s − 1.40·55-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  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 1.63T + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 + 0.846T + 19T^{2} \)
23 \( 1 + 4.10T + 23T^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 + 4.67T + 31T^{2} \)
37 \( 1 + 2.39T + 37T^{2} \)
41 \( 1 - 2.12T + 41T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 6.85T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 - 9.21T + 73T^{2} \)
79 \( 1 + 1.24T + 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + 5.06T + 89T^{2} \)
97 \( 1 - 3.83T + 97T^{2} \)
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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

−7.66516559598016217216174579220, −7.34620853162911517465696387124, −6.20232004797597284192148897075, −5.49263479342918118660919367603, −4.92761405155815159214178797395, −4.10487533723703180304150471971, −2.82854073017628264668778516102, −2.24103468480830672430347558309, −1.91320084651808532862733664029, 0, 1.91320084651808532862733664029, 2.24103468480830672430347558309, 2.82854073017628264668778516102, 4.10487533723703180304150471971, 4.92761405155815159214178797395, 5.49263479342918118660919367603, 6.20232004797597284192148897075, 7.34620853162911517465696387124, 7.66516559598016217216174579220

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