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  + 3.31·3-s + 1.75·5-s + 2.38·7-s + 8.00·9-s − 3.93·11-s − 0.976·13-s + 5.83·15-s − 2.10·17-s + 6.21·19-s + 7.89·21-s + 6.40·23-s − 1.90·25-s + 16.5·27-s + 0.867·29-s − 10.0·31-s − 13.0·33-s + 4.18·35-s + 0.819·37-s − 3.24·39-s + 4.02·41-s + 6.72·43-s + 14.0·45-s + 8.59·47-s − 1.32·49-s − 6.99·51-s + 2.10·53-s − 6.91·55-s + ⋯
L(s)  = 1  + 1.91·3-s + 0.786·5-s + 0.900·7-s + 2.66·9-s − 1.18·11-s − 0.270·13-s + 1.50·15-s − 0.511·17-s + 1.42·19-s + 1.72·21-s + 1.33·23-s − 0.381·25-s + 3.19·27-s + 0.161·29-s − 1.79·31-s − 2.26·33-s + 0.708·35-s + 0.134·37-s − 0.518·39-s + 0.628·41-s + 1.02·43-s + 2.09·45-s + 1.25·47-s − 0.189·49-s − 0.979·51-s + 0.289·53-s − 0.932·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.543841732$
$L(\frac12)$  $\approx$  $5.543841732$
$L(\frac{3}{2})$   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 \( 1 - 3.31T + 3T^{2} \)
5 \( 1 - 1.75T + 5T^{2} \)
7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 + 0.976T + 13T^{2} \)
17 \( 1 + 2.10T + 17T^{2} \)
19 \( 1 - 6.21T + 19T^{2} \)
23 \( 1 - 6.40T + 23T^{2} \)
29 \( 1 - 0.867T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 0.819T + 37T^{2} \)
41 \( 1 - 4.02T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 - 8.59T + 47T^{2} \)
53 \( 1 - 2.10T + 53T^{2} \)
59 \( 1 + 5.86T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 3.94T + 67T^{2} \)
71 \( 1 - 3.41T + 71T^{2} \)
73 \( 1 + 7.00T + 73T^{2} \)
79 \( 1 + 0.747T + 79T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 - 0.430T + 89T^{2} \)
97 \( 1 - 8.89T + 97T^{2} \)
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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

−7.996717940916762129546229360167, −7.49419635242817362217119003884, −7.17888740123052145320633647801, −5.81986284228883724556836269101, −5.07152819226287950919730799823, −4.40058009432006782331777422865, −3.37690609289512586287219698122, −2.68051782352774922817016673460, −2.08432760779280943114439613017, −1.26066097150147699733231382219, 1.26066097150147699733231382219, 2.08432760779280943114439613017, 2.68051782352774922817016673460, 3.37690609289512586287219698122, 4.40058009432006782331777422865, 5.07152819226287950919730799823, 5.81986284228883724556836269101, 7.17888740123052145320633647801, 7.49419635242817362217119003884, 7.996717940916762129546229360167

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