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  − 3.25·3-s − 1.89·5-s − 1.94·7-s + 7.59·9-s + 2.87·11-s − 5.69·13-s + 6.18·15-s + 0.264·17-s + 0.150·19-s + 6.32·21-s − 6.02·23-s − 1.39·25-s − 14.9·27-s + 6.65·29-s − 1.66·31-s − 9.34·33-s + 3.69·35-s + 11.9·37-s + 18.5·39-s − 4.78·41-s − 4.59·43-s − 14.4·45-s + 0.373·47-s − 3.22·49-s − 0.862·51-s − 3.24·53-s − 5.45·55-s + ⋯
L(s)  = 1  − 1.87·3-s − 0.849·5-s − 0.734·7-s + 2.53·9-s + 0.865·11-s − 1.58·13-s + 1.59·15-s + 0.0642·17-s + 0.0345·19-s + 1.38·21-s − 1.25·23-s − 0.278·25-s − 2.88·27-s + 1.23·29-s − 0.299·31-s − 1.62·33-s + 0.624·35-s + 1.96·37-s + 2.96·39-s − 0.746·41-s − 0.700·43-s − 2.15·45-s + 0.0544·47-s − 0.460·49-s − 0.120·51-s − 0.445·53-s − 0.735·55-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 \( 1 + 3.25T + 3T^{2} \)
5 \( 1 + 1.89T + 5T^{2} \)
7 \( 1 + 1.94T + 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
17 \( 1 - 0.264T + 17T^{2} \)
19 \( 1 - 0.150T + 19T^{2} \)
23 \( 1 + 6.02T + 23T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + 4.78T + 41T^{2} \)
43 \( 1 + 4.59T + 43T^{2} \)
47 \( 1 - 0.373T + 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 5.06T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 7.20T + 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + 4.67T + 97T^{2} \)
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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

−7.50181244504042660110156980192, −6.80016391439233627830115765282, −6.38525661026922682427744368957, −5.64375665564796751766490371930, −4.80879016178407106289715307316, −4.28634851456701222079407841496, −3.52600343443042439854113062432, −2.16233797755013592290606106911, −0.824210600073255000755724406671, 0, 0.824210600073255000755724406671, 2.16233797755013592290606106911, 3.52600343443042439854113062432, 4.28634851456701222079407841496, 4.80879016178407106289715307316, 5.64375665564796751766490371930, 6.38525661026922682427744368957, 6.80016391439233627830115765282, 7.50181244504042660110156980192

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