Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.00·3-s + 0.0492·5-s + 3.84·7-s + 1.02·9-s − 1.21·11-s + 1.27·13-s − 0.0987·15-s − 0.819·17-s − 1.24·19-s − 7.71·21-s + 1.37·23-s − 4.99·25-s + 3.95·27-s + 0.881·29-s − 1.56·31-s + 2.43·33-s + 0.189·35-s + 5.05·37-s − 2.55·39-s − 5.34·41-s − 9.30·43-s + 0.0506·45-s − 3.16·47-s + 7.76·49-s + 1.64·51-s + 3.52·53-s − 0.0597·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.0220·5-s + 1.45·7-s + 0.342·9-s − 0.366·11-s + 0.353·13-s − 0.0255·15-s − 0.198·17-s − 0.285·19-s − 1.68·21-s + 0.286·23-s − 0.999·25-s + 0.761·27-s + 0.163·29-s − 0.280·31-s + 0.424·33-s + 0.0319·35-s + 0.831·37-s − 0.409·39-s − 0.835·41-s − 1.41·43-s + 0.00754·45-s − 0.462·47-s + 1.10·49-s + 0.230·51-s + 0.484·53-s − 0.00806·55-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  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 2.00T + 3T^{2} \)
5 \( 1 - 0.0492T + 5T^{2} \)
7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + 0.819T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 - 0.881T + 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 - 5.05T + 37T^{2} \)
41 \( 1 + 5.34T + 41T^{2} \)
43 \( 1 + 9.30T + 43T^{2} \)
47 \( 1 + 3.16T + 47T^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 - 9.34T + 59T^{2} \)
61 \( 1 + 5.45T + 61T^{2} \)
67 \( 1 + 5.39T + 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 7.25T + 83T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−7.66668665793760541084469736306, −7.01751363007315737130062767648, −6.05614513915920132058064247822, −5.65944174856826396998565555653, −4.79762347630573080324487522575, −4.46633351103946708945580517998, −3.26886166532433195954497358492, −2.08158000365102589715407465434, −1.25477901657030875358561526320, 0, 1.25477901657030875358561526320, 2.08158000365102589715407465434, 3.26886166532433195954497358492, 4.46633351103946708945580517998, 4.79762347630573080324487522575, 5.65944174856826396998565555653, 6.05614513915920132058064247822, 7.01751363007315737130062767648, 7.66668665793760541084469736306

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