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  + 3.08·3-s − 3.00·5-s + 4.90·7-s + 6.52·9-s − 0.849·11-s + 3.14·13-s − 9.27·15-s + 0.172·17-s − 5.46·19-s + 15.1·21-s − 1.77·23-s + 4.03·25-s + 10.8·27-s + 4.39·29-s − 1.76·31-s − 2.62·33-s − 14.7·35-s + 7.37·37-s + 9.69·39-s + 7.48·41-s + 10.5·43-s − 19.6·45-s − 12.9·47-s + 17.0·49-s + 0.532·51-s + 4.69·53-s + 2.55·55-s + ⋯
L(s)  = 1  + 1.78·3-s − 1.34·5-s + 1.85·7-s + 2.17·9-s − 0.256·11-s + 0.871·13-s − 2.39·15-s + 0.0418·17-s − 1.25·19-s + 3.30·21-s − 0.369·23-s + 0.806·25-s + 2.09·27-s + 0.815·29-s − 0.316·31-s − 0.456·33-s − 2.49·35-s + 1.21·37-s + 1.55·39-s + 1.16·41-s + 1.61·43-s − 2.92·45-s − 1.88·47-s + 2.44·49-s + 0.0746·51-s + 0.644·53-s + 0.344·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.153672936$
$L(\frac12)$  $\approx$  $4.153672936$
$L(\frac{3}{2})$   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 \( 1 - 3.08T + 3T^{2} \)
5 \( 1 + 3.00T + 5T^{2} \)
7 \( 1 - 4.90T + 7T^{2} \)
11 \( 1 + 0.849T + 11T^{2} \)
13 \( 1 - 3.14T + 13T^{2} \)
17 \( 1 - 0.172T + 17T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 - 4.39T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 - 7.37T + 37T^{2} \)
41 \( 1 - 7.48T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 4.69T + 53T^{2} \)
59 \( 1 - 6.35T + 59T^{2} \)
61 \( 1 - 7.33T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 4.18T + 71T^{2} \)
73 \( 1 - 3.73T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 0.877T + 83T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−8.037573371948077856408735596146, −7.77433834982207011994777508602, −7.14594147893735175712908783527, −5.99642819384790352671517221309, −4.71734788941075346501871271486, −4.24184821618978066192615319868, −3.79140332668595695325065230841, −2.74690203935553951623502157222, −2.01731761645240002536306957534, −1.05623125550272660015074438502, 1.05623125550272660015074438502, 2.01731761645240002536306957534, 2.74690203935553951623502157222, 3.79140332668595695325065230841, 4.24184821618978066192615319868, 4.71734788941075346501871271486, 5.99642819384790352671517221309, 7.14594147893735175712908783527, 7.77433834982207011994777508602, 8.037573371948077856408735596146

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