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  + 0.320·3-s + 1.20·5-s + 2.62·7-s − 2.89·9-s + 1.62·11-s − 3.31·13-s + 0.386·15-s − 2.66·17-s + 4.21·19-s + 0.842·21-s − 1.23·23-s − 3.54·25-s − 1.89·27-s − 9.10·29-s + 5.03·31-s + 0.521·33-s + 3.16·35-s − 8.64·37-s − 1.06·39-s − 6.06·41-s − 5.22·43-s − 3.49·45-s − 2.33·47-s − 0.109·49-s − 0.855·51-s + 1.51·53-s + 1.96·55-s + ⋯
L(s)  = 1  + 0.185·3-s + 0.539·5-s + 0.992·7-s − 0.965·9-s + 0.490·11-s − 0.920·13-s + 0.0998·15-s − 0.646·17-s + 0.967·19-s + 0.183·21-s − 0.257·23-s − 0.709·25-s − 0.364·27-s − 1.69·29-s + 0.904·31-s + 0.0908·33-s + 0.534·35-s − 1.42·37-s − 0.170·39-s − 0.946·41-s − 0.797·43-s − 0.520·45-s − 0.340·47-s − 0.0156·49-s − 0.119·51-s + 0.208·53-s + 0.264·55-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 \( 1 - 0.320T + 3T^{2} \)
5 \( 1 - 1.20T + 5T^{2} \)
7 \( 1 - 2.62T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + 9.10T + 29T^{2} \)
31 \( 1 - 5.03T + 31T^{2} \)
37 \( 1 + 8.64T + 37T^{2} \)
41 \( 1 + 6.06T + 41T^{2} \)
43 \( 1 + 5.22T + 43T^{2} \)
47 \( 1 + 2.33T + 47T^{2} \)
53 \( 1 - 1.51T + 53T^{2} \)
59 \( 1 - 9.59T + 59T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 + 3.57T + 73T^{2} \)
79 \( 1 - 2.62T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−7.73328850877262210874340291130, −7.12135279694522840220953722578, −6.25562585083179343701817182585, −5.40935956851685681401418974131, −5.06083894565712327646450283025, −4.04521902746057847781002359487, −3.15919037745391627504718667257, −2.19264603083387780811660864545, −1.57710875476829425541941700174, 0, 1.57710875476829425541941700174, 2.19264603083387780811660864545, 3.15919037745391627504718667257, 4.04521902746057847781002359487, 5.06083894565712327646450283025, 5.40935956851685681401418974131, 6.25562585083179343701817182585, 7.12135279694522840220953722578, 7.73328850877262210874340291130

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