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  − 0.447·3-s + 2.48·5-s + 3.20·7-s − 2.79·9-s − 0.332·11-s − 4.86·13-s − 1.11·15-s − 4.61·17-s − 7.39·19-s − 1.43·21-s + 5.99·23-s + 1.17·25-s + 2.59·27-s + 8.23·29-s + 7.16·31-s + 0.148·33-s + 7.97·35-s − 5.58·37-s + 2.17·39-s − 3.18·41-s + 3.29·43-s − 6.95·45-s − 9.99·47-s + 3.29·49-s + 2.06·51-s − 2.53·53-s − 0.825·55-s + ⋯
L(s)  = 1  − 0.258·3-s + 1.11·5-s + 1.21·7-s − 0.933·9-s − 0.100·11-s − 1.34·13-s − 0.287·15-s − 1.11·17-s − 1.69·19-s − 0.313·21-s + 1.25·23-s + 0.234·25-s + 0.499·27-s + 1.52·29-s + 1.28·31-s + 0.0259·33-s + 1.34·35-s − 0.918·37-s + 0.348·39-s − 0.496·41-s + 0.501·43-s − 1.03·45-s − 1.45·47-s + 0.471·49-s + 0.289·51-s − 0.348·53-s − 0.111·55-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 \( 1 + 0.447T + 3T^{2} \)
5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 + 0.332T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 + 4.61T + 17T^{2} \)
19 \( 1 + 7.39T + 19T^{2} \)
23 \( 1 - 5.99T + 23T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 - 7.16T + 31T^{2} \)
37 \( 1 + 5.58T + 37T^{2} \)
41 \( 1 + 3.18T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + 9.99T + 47T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + 5.30T + 59T^{2} \)
61 \( 1 - 8.55T + 61T^{2} \)
67 \( 1 + 5.87T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 2.82T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 6.60T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−7.85463433590548407715055299280, −6.73104498828776879946658930939, −6.43144912492169792698857979632, −5.46644338983165332335554654669, −4.82224286934525501632476850567, −4.49076505997804970407364464655, −2.83720298143620388366473894001, −2.35497293166485205249051117970, −1.48864353871027668672471374630, 0, 1.48864353871027668672471374630, 2.35497293166485205249051117970, 2.83720298143620388366473894001, 4.49076505997804970407364464655, 4.82224286934525501632476850567, 5.46644338983165332335554654669, 6.43144912492169792698857979632, 6.73104498828776879946658930939, 7.85463433590548407715055299280

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