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  + 0.765·3-s + 3.23·5-s − 3.39·7-s − 2.41·9-s − 5.84·11-s − 4.58·13-s + 2.47·15-s + 2.17·17-s + 0.556·19-s − 2.59·21-s + 6.51·23-s + 5.47·25-s − 4.14·27-s + 8.72·29-s + 3.80·31-s − 4.47·33-s − 10.9·35-s + 8.20·37-s − 3.50·39-s + 4.82·41-s + 11.5·43-s − 7.81·45-s − 5.50·47-s + 4.51·49-s + 1.66·51-s − 13.2·53-s − 18.9·55-s + ⋯
L(s)  = 1  + 0.441·3-s + 1.44·5-s − 1.28·7-s − 0.804·9-s − 1.76·11-s − 1.27·13-s + 0.639·15-s + 0.528·17-s + 0.127·19-s − 0.566·21-s + 1.35·23-s + 1.09·25-s − 0.797·27-s + 1.62·29-s + 0.683·31-s − 0.779·33-s − 1.85·35-s + 1.34·37-s − 0.561·39-s + 0.753·41-s + 1.76·43-s − 1.16·45-s − 0.803·47-s + 0.645·49-s + 0.233·51-s − 1.81·53-s − 2.55·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.894211652$
$L(\frac12)$  $\approx$  $1.894211652$
$L(\frac{3}{2})$   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 \( 1 - 0.765T + 3T^{2} \)
5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 3.39T + 7T^{2} \)
11 \( 1 + 5.84T + 11T^{2} \)
13 \( 1 + 4.58T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 - 0.556T + 19T^{2} \)
23 \( 1 - 6.51T + 23T^{2} \)
29 \( 1 - 8.72T + 29T^{2} \)
31 \( 1 - 3.80T + 31T^{2} \)
37 \( 1 - 8.20T + 37T^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 5.50T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 - 6.71T + 67T^{2} \)
71 \( 1 - 4.59T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + 7.64T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 1.70T + 97T^{2} \)
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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

−7.944542380122212491925155850870, −7.53796960734747315972741498001, −6.39668039177543473476249994538, −6.08558101151894489309667718740, −5.20315026936818020786432049009, −4.74953553805373051662102370700, −3.00672513911367113067418268466, −2.89591101297046514913067250174, −2.25387600873131878197222118863, −0.66519419146197218027975405831, 0.66519419146197218027975405831, 2.25387600873131878197222118863, 2.89591101297046514913067250174, 3.00672513911367113067418268466, 4.74953553805373051662102370700, 5.20315026936818020786432049009, 6.08558101151894489309667718740, 6.39668039177543473476249994538, 7.53796960734747315972741498001, 7.944542380122212491925155850870

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