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  + 2.15·3-s + 1.11·5-s − 1.03·7-s + 1.65·9-s + 2.85·11-s + 3.74·13-s + 2.40·15-s + 5.19·17-s + 1.79·19-s − 2.23·21-s + 6.06·23-s − 3.75·25-s − 2.90·27-s + 0.462·29-s + 1.21·31-s + 6.14·33-s − 1.15·35-s − 7.34·37-s + 8.06·39-s + 6.52·41-s + 2.05·43-s + 1.84·45-s − 5.41·47-s − 5.92·49-s + 11.2·51-s + 2.63·53-s + 3.18·55-s + ⋯
L(s)  = 1  + 1.24·3-s + 0.499·5-s − 0.391·7-s + 0.550·9-s + 0.859·11-s + 1.03·13-s + 0.621·15-s + 1.26·17-s + 0.411·19-s − 0.487·21-s + 1.26·23-s − 0.750·25-s − 0.559·27-s + 0.0859·29-s + 0.217·31-s + 1.07·33-s − 0.195·35-s − 1.20·37-s + 1.29·39-s + 1.01·41-s + 0.314·43-s + 0.274·45-s − 0.790·47-s − 0.846·49-s + 1.56·51-s + 0.362·53-s + 0.429·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.009624376$
$L(\frac12)$  $\approx$  $4.009624376$
$L(\frac{3}{2})$   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 \( 1 - 2.15T + 3T^{2} \)
5 \( 1 - 1.11T + 5T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
13 \( 1 - 3.74T + 13T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 - 1.79T + 19T^{2} \)
23 \( 1 - 6.06T + 23T^{2} \)
29 \( 1 - 0.462T + 29T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + 7.34T + 37T^{2} \)
41 \( 1 - 6.52T + 41T^{2} \)
43 \( 1 - 2.05T + 43T^{2} \)
47 \( 1 + 5.41T + 47T^{2} \)
53 \( 1 - 2.63T + 53T^{2} \)
59 \( 1 + 1.09T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 2.35T + 71T^{2} \)
73 \( 1 + 6.45T + 73T^{2} \)
79 \( 1 - 0.844T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 0.307T + 89T^{2} \)
97 \( 1 + 5.44T + 97T^{2} \)
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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

−8.182670715756448246634683967910, −7.48879063194255537346088229324, −6.73042881408957125404397568958, −5.97226668033295478477345825750, −5.32189957380200142509118587218, −4.14890886508049147281738041184, −3.41377775538902081064187764220, −3.00322228738002342188200104261, −1.87922848331300222869832455541, −1.07679566745215518805449622944, 1.07679566745215518805449622944, 1.87922848331300222869832455541, 3.00322228738002342188200104261, 3.41377775538902081064187764220, 4.14890886508049147281738041184, 5.32189957380200142509118587218, 5.97226668033295478477345825750, 6.73042881408957125404397568958, 7.48879063194255537346088229324, 8.182670715756448246634683967910

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