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  + 2.26·3-s − 2.03·5-s + 2.84·7-s + 2.12·9-s + 5.94·11-s + 3.06·13-s − 4.59·15-s + 5.90·17-s + 2.58·19-s + 6.44·21-s + 1.10·23-s − 0.874·25-s − 1.98·27-s + 6.71·29-s − 6.73·31-s + 13.4·33-s − 5.78·35-s + 4.99·37-s + 6.93·39-s − 8.52·41-s − 5.24·43-s − 4.31·45-s + 10.6·47-s + 1.11·49-s + 13.3·51-s − 10.2·53-s − 12.0·55-s + ⋯
L(s)  = 1  + 1.30·3-s − 0.908·5-s + 1.07·7-s + 0.707·9-s + 1.79·11-s + 0.849·13-s − 1.18·15-s + 1.43·17-s + 0.592·19-s + 1.40·21-s + 0.230·23-s − 0.174·25-s − 0.381·27-s + 1.24·29-s − 1.21·31-s + 2.34·33-s − 0.978·35-s + 0.821·37-s + 1.11·39-s − 1.33·41-s − 0.800·43-s − 0.642·45-s + 1.54·47-s + 0.159·49-s + 1.87·51-s − 1.41·53-s − 1.62·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.992469734$
$L(\frac12)$  $\approx$  $3.992469734$
$L(\frac{3}{2})$   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 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 + 2.03T + 5T^{2} \)
7 \( 1 - 2.84T + 7T^{2} \)
11 \( 1 - 5.94T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 - 5.90T + 17T^{2} \)
19 \( 1 - 2.58T + 19T^{2} \)
23 \( 1 - 1.10T + 23T^{2} \)
29 \( 1 - 6.71T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 + 8.52T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 2.18T + 59T^{2} \)
61 \( 1 + 7.25T + 61T^{2} \)
67 \( 1 - 7.02T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 8.01T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 4.69T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−8.155495372658900225438292402841, −7.64004747953091691688655223314, −6.92175301474011934726355183599, −6.00026856279709790455650393694, −5.03547161030352897509271779548, −4.09692399321285102090722660848, −3.64441939649042506250741297390, −3.04323106057879200473367136545, −1.71351572822590095598360444283, −1.13436945614072573785820928697, 1.13436945614072573785820928697, 1.71351572822590095598360444283, 3.04323106057879200473367136545, 3.64441939649042506250741297390, 4.09692399321285102090722660848, 5.03547161030352897509271779548, 6.00026856279709790455650393694, 6.92175301474011934726355183599, 7.64004747953091691688655223314, 8.155495372658900225438292402841

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