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  − 2.99·3-s + 0.770·5-s + 1.05·7-s + 5.99·9-s + 0.959·11-s + 2.31·13-s − 2.30·15-s + 0.658·17-s + 0.523·19-s − 3.15·21-s + 2.38·23-s − 4.40·25-s − 8.96·27-s − 3.71·29-s − 10.3·31-s − 2.87·33-s + 0.810·35-s + 3.08·37-s − 6.92·39-s + 5.97·41-s − 5.82·43-s + 4.61·45-s − 8.50·47-s − 5.89·49-s − 1.97·51-s − 0.661·53-s + 0.738·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.344·5-s + 0.397·7-s + 1.99·9-s + 0.289·11-s + 0.640·13-s − 0.596·15-s + 0.159·17-s + 0.120·19-s − 0.688·21-s + 0.497·23-s − 0.881·25-s − 1.72·27-s − 0.689·29-s − 1.86·31-s − 0.500·33-s + 0.137·35-s + 0.506·37-s − 1.10·39-s + 0.932·41-s − 0.888·43-s + 0.687·45-s − 1.24·47-s − 0.841·49-s − 0.276·51-s − 0.0908·53-s + 0.0995·55-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 \( 1 + 2.99T + 3T^{2} \)
5 \( 1 - 0.770T + 5T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 - 0.959T + 11T^{2} \)
13 \( 1 - 2.31T + 13T^{2} \)
17 \( 1 - 0.658T + 17T^{2} \)
19 \( 1 - 0.523T + 19T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 + 3.71T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 3.08T + 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 + 8.50T + 47T^{2} \)
53 \( 1 + 0.661T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 5.54T + 61T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 - 3.51T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 0.0967T + 79T^{2} \)
83 \( 1 - 2.45T + 83T^{2} \)
89 \( 1 + 8.07T + 89T^{2} \)
97 \( 1 + 6.95T + 97T^{2} \)
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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

−7.56359222203011952762347296081, −6.78523188545625569597837887165, −6.22631059561146169706906234687, −5.53910434221349377171486036006, −5.12030295109967540762217683337, −4.23257249089369000842270142531, −3.46100884301685122350240044339, −1.93133905934819423890414918950, −1.21037764448722033468858758853, 0, 1.21037764448722033468858758853, 1.93133905934819423890414918950, 3.46100884301685122350240044339, 4.23257249089369000842270142531, 5.12030295109967540762217683337, 5.53910434221349377171486036006, 6.22631059561146169706906234687, 6.78523188545625569597837887165, 7.56359222203011952762347296081

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