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  + 2.13·3-s − 1.12·5-s − 4.02·7-s + 1.54·9-s − 5.57·11-s + 0.335·13-s − 2.39·15-s + 4.79·17-s + 1.73·19-s − 8.58·21-s − 3.07·23-s − 3.74·25-s − 3.09·27-s + 6.66·29-s − 3.51·31-s − 11.8·33-s + 4.51·35-s + 5.13·37-s + 0.715·39-s + 4.64·41-s − 9.81·43-s − 1.73·45-s + 11.9·47-s + 9.20·49-s + 10.2·51-s + 11.4·53-s + 6.25·55-s + ⋯
L(s)  = 1  + 1.23·3-s − 0.501·5-s − 1.52·7-s + 0.515·9-s − 1.67·11-s + 0.0930·13-s − 0.618·15-s + 1.16·17-s + 0.397·19-s − 1.87·21-s − 0.641·23-s − 0.748·25-s − 0.596·27-s + 1.23·29-s − 0.630·31-s − 2.06·33-s + 0.763·35-s + 0.844·37-s + 0.114·39-s + 0.725·41-s − 1.49·43-s − 0.258·45-s + 1.73·47-s + 1.31·49-s + 1.43·51-s + 1.57·53-s + 0.843·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.641789706$
$L(\frac12)$  $\approx$  $1.641789706$
$L(\frac{3}{2})$   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 \( 1 - 2.13T + 3T^{2} \)
5 \( 1 + 1.12T + 5T^{2} \)
7 \( 1 + 4.02T + 7T^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 - 0.335T + 13T^{2} \)
17 \( 1 - 4.79T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 + 3.07T + 23T^{2} \)
29 \( 1 - 6.66T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 - 5.13T + 37T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 + 9.81T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 2.35T + 59T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 - 6.65T + 73T^{2} \)
79 \( 1 + 8.62T + 79T^{2} \)
83 \( 1 + 1.96T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 5.97T + 97T^{2} \)
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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

−7.951120163870238061496879958801, −7.66809690654240144006804704322, −6.87055951703269685245243252571, −5.87198385194703990315672061792, −5.34072454614511773641796901583, −4.06717641190508177438372023644, −3.48468940752082730912133568481, −2.84510253744552249006995020791, −2.28926043117533003493671888488, −0.59525813923847470156447925414, 0.59525813923847470156447925414, 2.28926043117533003493671888488, 2.84510253744552249006995020791, 3.48468940752082730912133568481, 4.06717641190508177438372023644, 5.34072454614511773641796901583, 5.87198385194703990315672061792, 6.87055951703269685245243252571, 7.66809690654240144006804704322, 7.951120163870238061496879958801

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