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  − 0.691·3-s − 1.88·5-s + 4.98·7-s − 2.52·9-s − 4.24·11-s + 0.255·13-s + 1.30·15-s − 8.15·17-s − 0.0286·19-s − 3.44·21-s + 6.59·23-s − 1.42·25-s + 3.81·27-s − 2.26·29-s − 4.18·31-s + 2.93·33-s − 9.42·35-s − 4.27·37-s − 0.176·39-s − 7.77·41-s + 11.8·43-s + 4.76·45-s + 0.0979·47-s + 17.8·49-s + 5.64·51-s + 2.31·53-s + 8.01·55-s + ⋯
L(s)  = 1  − 0.399·3-s − 0.845·5-s + 1.88·7-s − 0.840·9-s − 1.27·11-s + 0.0708·13-s + 0.337·15-s − 1.97·17-s − 0.00656·19-s − 0.752·21-s + 1.37·23-s − 0.285·25-s + 0.734·27-s − 0.420·29-s − 0.752·31-s + 0.510·33-s − 1.59·35-s − 0.702·37-s − 0.0283·39-s − 1.21·41-s + 1.81·43-s + 0.710·45-s + 0.0142·47-s + 2.55·49-s + 0.790·51-s + 0.317·53-s + 1.08·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.031138085$
$L(\frac12)$  $\approx$  $1.031138085$
$L(\frac{3}{2})$   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 \( 1 + 0.691T + 3T^{2} \)
5 \( 1 + 1.88T + 5T^{2} \)
7 \( 1 - 4.98T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 0.255T + 13T^{2} \)
17 \( 1 + 8.15T + 17T^{2} \)
19 \( 1 + 0.0286T + 19T^{2} \)
23 \( 1 - 6.59T + 23T^{2} \)
29 \( 1 + 2.26T + 29T^{2} \)
31 \( 1 + 4.18T + 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 + 7.77T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 0.0979T + 47T^{2} \)
53 \( 1 - 2.31T + 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 - 3.40T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 5.63T + 73T^{2} \)
79 \( 1 - 7.59T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 2.68T + 89T^{2} \)
97 \( 1 + 1.68T + 97T^{2} \)
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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

−8.090666301348933318266065170244, −7.47080389512525880224785751079, −6.84457045847887379853484404030, −5.68888538324020049743000196592, −5.11748100624832775058932679615, −4.64948998916058644350265308402, −3.80727587377349149453166138293, −2.62902562309712839275191106134, −1.93131438869101326466876158288, −0.52635149899877375608417264097, 0.52635149899877375608417264097, 1.93131438869101326466876158288, 2.62902562309712839275191106134, 3.80727587377349149453166138293, 4.64948998916058644350265308402, 5.11748100624832775058932679615, 5.68888538324020049743000196592, 6.84457045847887379853484404030, 7.47080389512525880224785751079, 8.090666301348933318266065170244

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