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  − 1.33·3-s + 3.94·5-s + 0.220·7-s − 1.20·9-s + 1.87·11-s + 3.17·13-s − 5.28·15-s + 7.01·17-s + 5.00·19-s − 0.295·21-s + 1.70·23-s + 10.5·25-s + 5.63·27-s + 1.30·29-s − 4.27·31-s − 2.50·33-s + 0.871·35-s + 3.57·37-s − 4.25·39-s + 4.53·41-s − 3.45·43-s − 4.77·45-s + 1.75·47-s − 6.95·49-s − 9.38·51-s − 6.18·53-s + 7.38·55-s + ⋯
L(s)  = 1  − 0.772·3-s + 1.76·5-s + 0.0834·7-s − 0.402·9-s + 0.564·11-s + 0.880·13-s − 1.36·15-s + 1.70·17-s + 1.14·19-s − 0.0644·21-s + 0.355·23-s + 2.11·25-s + 1.08·27-s + 0.242·29-s − 0.767·31-s − 0.436·33-s + 0.147·35-s + 0.587·37-s − 0.680·39-s + 0.708·41-s − 0.527·43-s − 0.711·45-s + 0.255·47-s − 0.993·49-s − 1.31·51-s − 0.850·53-s + 0.996·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.666385503$
$L(\frac12)$  $\approx$  $2.666385503$
$L(\frac{3}{2})$   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 \( 1 + 1.33T + 3T^{2} \)
5 \( 1 - 3.94T + 5T^{2} \)
7 \( 1 - 0.220T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 - 7.01T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
23 \( 1 - 1.70T + 23T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 + 4.27T + 31T^{2} \)
37 \( 1 - 3.57T + 37T^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 8.37T + 61T^{2} \)
67 \( 1 + 3.87T + 67T^{2} \)
71 \( 1 - 3.47T + 71T^{2} \)
73 \( 1 + 6.72T + 73T^{2} \)
79 \( 1 + 6.08T + 79T^{2} \)
83 \( 1 + 6.03T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 0.167T + 97T^{2} \)
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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

−8.107391090498764424685291646998, −7.20449705328941860853544195278, −6.32720607604378554474612604835, −5.92099128969478197866002588263, −5.43736072142244271583253822574, −4.80840696652427336192047815071, −3.47116527788634055978105022408, −2.81041731168986227285852003688, −1.55467869711872911389801039227, −1.01702705051917776136472364941, 1.01702705051917776136472364941, 1.55467869711872911389801039227, 2.81041731168986227285852003688, 3.47116527788634055978105022408, 4.80840696652427336192047815071, 5.43736072142244271583253822574, 5.92099128969478197866002588263, 6.32720607604378554474612604835, 7.20449705328941860853544195278, 8.107391090498764424685291646998

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