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  − 1.34·3-s + 0.0749·5-s − 4.25·7-s − 1.18·9-s + 1.12·11-s − 0.485·13-s − 0.100·15-s + 6.54·17-s − 3.86·19-s + 5.72·21-s − 8.49·23-s − 4.99·25-s + 5.63·27-s − 1.99·29-s − 6.58·31-s − 1.51·33-s − 0.318·35-s + 5.71·37-s + 0.654·39-s + 4.59·41-s − 6.88·43-s − 0.0887·45-s + 2.43·47-s + 11.0·49-s − 8.81·51-s − 10.3·53-s + 0.0840·55-s + ⋯
L(s)  = 1  − 0.777·3-s + 0.0334·5-s − 1.60·7-s − 0.394·9-s + 0.338·11-s − 0.134·13-s − 0.0260·15-s + 1.58·17-s − 0.886·19-s + 1.25·21-s − 1.77·23-s − 0.998·25-s + 1.08·27-s − 0.371·29-s − 1.18·31-s − 0.263·33-s − 0.0538·35-s + 0.939·37-s + 0.104·39-s + 0.717·41-s − 1.05·43-s − 0.0132·45-s + 0.355·47-s + 1.58·49-s − 1.23·51-s − 1.42·53-s + 0.0113·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4438949164$
$L(\frac12)$  $\approx$  $0.4438949164$
$L(\frac{3}{2})$   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 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 - 0.0749T + 5T^{2} \)
7 \( 1 + 4.25T + 7T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 + 0.485T + 13T^{2} \)
17 \( 1 - 6.54T + 17T^{2} \)
19 \( 1 + 3.86T + 19T^{2} \)
23 \( 1 + 8.49T + 23T^{2} \)
29 \( 1 + 1.99T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 - 5.71T + 37T^{2} \)
41 \( 1 - 4.59T + 41T^{2} \)
43 \( 1 + 6.88T + 43T^{2} \)
47 \( 1 - 2.43T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 5.14T + 59T^{2} \)
61 \( 1 + 5.48T + 61T^{2} \)
67 \( 1 + 8.24T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + 1.73T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 6.39T + 89T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−7.951349624332411554869625743361, −7.33909748494736757747334160509, −6.31120100909483601731431718520, −6.04156849273486791924642614436, −5.55648304038563484405130903655, −4.38137145247104761035350467970, −3.60946202518809770658705053761, −2.94599164104428814386303528833, −1.78898968257527753857284283055, −0.34923828408716883700221387056, 0.34923828408716883700221387056, 1.78898968257527753857284283055, 2.94599164104428814386303528833, 3.60946202518809770658705053761, 4.38137145247104761035350467970, 5.55648304038563484405130903655, 6.04156849273486791924642614436, 6.31120100909483601731431718520, 7.33909748494736757747334160509, 7.951349624332411554869625743361

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