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  − 2.41·3-s + 0.877·5-s + 2.89·7-s + 2.84·9-s + 1.81·11-s − 3.69·13-s − 2.12·15-s − 5.52·17-s − 3.24·19-s − 6.99·21-s − 2.45·23-s − 4.23·25-s + 0.369·27-s + 2.86·29-s − 7.81·31-s − 4.38·33-s + 2.53·35-s + 4.93·37-s + 8.94·39-s + 12.1·41-s + 6.06·43-s + 2.49·45-s + 4.72·47-s + 1.35·49-s + 13.3·51-s − 1.39·53-s + 1.59·55-s + ⋯
L(s)  = 1  − 1.39·3-s + 0.392·5-s + 1.09·7-s + 0.948·9-s + 0.546·11-s − 1.02·13-s − 0.547·15-s − 1.33·17-s − 0.743·19-s − 1.52·21-s − 0.512·23-s − 0.846·25-s + 0.0712·27-s + 0.532·29-s − 1.40·31-s − 0.762·33-s + 0.428·35-s + 0.810·37-s + 1.43·39-s + 1.89·41-s + 0.924·43-s + 0.372·45-s + 0.689·47-s + 0.194·49-s + 1.87·51-s − 0.191·53-s + 0.214·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.060080165$
$L(\frac12)$  $\approx$  $1.060080165$
$L(\frac{3}{2})$   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 \( 1 + 2.41T + 3T^{2} \)
5 \( 1 - 0.877T + 5T^{2} \)
7 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 + 3.24T + 19T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + 7.81T + 31T^{2} \)
37 \( 1 - 4.93T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 6.06T + 43T^{2} \)
47 \( 1 - 4.72T + 47T^{2} \)
53 \( 1 + 1.39T + 53T^{2} \)
59 \( 1 + 8.45T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 8.83T + 67T^{2} \)
71 \( 1 + 3.13T + 71T^{2} \)
73 \( 1 - 9.61T + 73T^{2} \)
79 \( 1 - 0.251T + 79T^{2} \)
83 \( 1 - 1.82T + 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.897695479893587416200040533843, −7.30275726561917238259364508417, −6.45538362131643371947183161483, −5.95036453995006520391470918803, −5.27056596531922236513435770564, −4.50020216978765695543838520385, −4.13459012363723000182651564239, −2.44746532013663791770704567248, −1.80316539406141680868675941067, −0.57638194209339431069289446824, 0.57638194209339431069289446824, 1.80316539406141680868675941067, 2.44746532013663791770704567248, 4.13459012363723000182651564239, 4.50020216978765695543838520385, 5.27056596531922236513435770564, 5.95036453995006520391470918803, 6.45538362131643371947183161483, 7.30275726561917238259364508417, 7.897695479893587416200040533843

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