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  − 1.09·3-s − 2.74·5-s + 2.16·7-s − 1.79·9-s − 1.58·11-s − 1.73·13-s + 3.01·15-s + 2.86·17-s + 1.23·19-s − 2.38·21-s + 1.19·23-s + 2.54·25-s + 5.26·27-s − 0.223·29-s − 1.89·31-s + 1.74·33-s − 5.95·35-s + 8.49·37-s + 1.90·39-s − 10.1·41-s + 6.87·43-s + 4.91·45-s − 2.10·47-s − 2.29·49-s − 3.15·51-s + 10.0·53-s + 4.35·55-s + ⋯
L(s)  = 1  − 0.634·3-s − 1.22·5-s + 0.819·7-s − 0.597·9-s − 0.477·11-s − 0.480·13-s + 0.779·15-s + 0.695·17-s + 0.282·19-s − 0.520·21-s + 0.248·23-s + 0.508·25-s + 1.01·27-s − 0.0415·29-s − 0.340·31-s + 0.303·33-s − 1.00·35-s + 1.39·37-s + 0.304·39-s − 1.58·41-s + 1.04·43-s + 0.733·45-s − 0.307·47-s − 0.327·49-s − 0.441·51-s + 1.37·53-s + 0.586·55-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 \( 1 + 1.09T + 3T^{2} \)
5 \( 1 + 2.74T + 5T^{2} \)
7 \( 1 - 2.16T + 7T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 - 2.86T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 - 1.19T + 23T^{2} \)
29 \( 1 + 0.223T + 29T^{2} \)
31 \( 1 + 1.89T + 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 6.87T + 43T^{2} \)
47 \( 1 + 2.10T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 3.43T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 6.11T + 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−7.79296243554769695699118259613, −7.18816797678840300380128296574, −6.26771335093787151413558325879, −5.37159001436539329987011041705, −4.97761190988794405926426823040, −4.13259959603152243406268006337, −3.29860400693772455219996368660, −2.40179445599813705015390854146, −1.03885552454624491064135648166, 0, 1.03885552454624491064135648166, 2.40179445599813705015390854146, 3.29860400693772455219996368660, 4.13259959603152243406268006337, 4.97761190988794405926426823040, 5.37159001436539329987011041705, 6.26771335093787151413558325879, 7.18816797678840300380128296574, 7.79296243554769695699118259613

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