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  − 1.56·3-s + 2.55·5-s − 2.27·7-s − 0.552·9-s − 5.49·11-s + 6.01·13-s − 3.99·15-s − 3.09·17-s − 4.77·19-s + 3.56·21-s + 7.81·23-s + 1.52·25-s + 5.55·27-s + 10.1·29-s − 9.04·31-s + 8.59·33-s − 5.81·35-s − 7.43·37-s − 9.41·39-s + 2.58·41-s − 8.81·43-s − 1.41·45-s − 0.566·47-s − 1.81·49-s + 4.83·51-s − 1.06·53-s − 14.0·55-s + ⋯
L(s)  = 1  − 0.903·3-s + 1.14·5-s − 0.860·7-s − 0.184·9-s − 1.65·11-s + 1.66·13-s − 1.03·15-s − 0.749·17-s − 1.09·19-s + 0.777·21-s + 1.62·23-s + 0.304·25-s + 1.06·27-s + 1.88·29-s − 1.62·31-s + 1.49·33-s − 0.982·35-s − 1.22·37-s − 1.50·39-s + 0.403·41-s − 1.34·43-s − 0.210·45-s − 0.0826·47-s − 0.259·49-s + 0.677·51-s − 0.145·53-s − 1.89·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.051042649$
$L(\frac12)$  $\approx$  $1.051042649$
$L(\frac{3}{2})$   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 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 2.55T + 5T^{2} \)
7 \( 1 + 2.27T + 7T^{2} \)
11 \( 1 + 5.49T + 11T^{2} \)
13 \( 1 - 6.01T + 13T^{2} \)
17 \( 1 + 3.09T + 17T^{2} \)
19 \( 1 + 4.77T + 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 9.04T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 + 0.566T + 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 - 8.80T + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 9.02T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 4.91T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 - 6.37T + 89T^{2} \)
97 \( 1 - 2.29T + 97T^{2} \)
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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

−8.323165699865117075729671157178, −7.02184493326072143568924302456, −6.42923298791027017299707806829, −6.05205883763894556237051925791, −5.28068437074344260110471136249, −4.82862034603777166277773950370, −3.49227781795512602486358022249, −2.77132751661341420627920692928, −1.84071054503518316563396145349, −0.54594930717050332536777160166, 0.54594930717050332536777160166, 1.84071054503518316563396145349, 2.77132751661341420627920692928, 3.49227781795512602486358022249, 4.82862034603777166277773950370, 5.28068437074344260110471136249, 6.05205883763894556237051925791, 6.42923298791027017299707806829, 7.02184493326072143568924302456, 8.323165699865117075729671157178

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