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  − 1.52·3-s + 0.276·5-s + 2.37·7-s − 0.663·9-s − 0.982·11-s − 5.12·13-s − 0.422·15-s + 7.66·17-s + 4.67·19-s − 3.63·21-s − 7.09·23-s − 4.92·25-s + 5.59·27-s + 2.66·29-s − 5.74·31-s + 1.50·33-s + 0.657·35-s − 0.273·37-s + 7.83·39-s − 4.12·41-s + 7.07·43-s − 0.183·45-s + 12.1·47-s − 1.33·49-s − 11.7·51-s − 0.726·53-s − 0.271·55-s + ⋯
L(s)  = 1  − 0.882·3-s + 0.123·5-s + 0.899·7-s − 0.221·9-s − 0.296·11-s − 1.42·13-s − 0.109·15-s + 1.85·17-s + 1.07·19-s − 0.793·21-s − 1.47·23-s − 0.984·25-s + 1.07·27-s + 0.494·29-s − 1.03·31-s + 0.261·33-s + 0.111·35-s − 0.0449·37-s + 1.25·39-s − 0.644·41-s + 1.07·43-s − 0.0273·45-s + 1.77·47-s − 0.191·49-s − 1.64·51-s − 0.0998·53-s − 0.0366·55-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 \( 1 + 1.52T + 3T^{2} \)
5 \( 1 - 0.276T + 5T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 + 0.982T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 - 7.66T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + 7.09T + 23T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 + 5.74T + 31T^{2} \)
37 \( 1 + 0.273T + 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 0.726T + 53T^{2} \)
59 \( 1 + 6.80T + 59T^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 - 6.63T + 97T^{2} \)
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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

−7.61946981076169765809075698079, −7.24319797421775232631903345366, −5.94110041103914456544383732185, −5.60418548234518930767719640750, −5.06870680766319278913015395048, −4.24222845533843803943618371786, −3.18903907366865131288320582938, −2.25060759300894240501711562559, −1.21132753789727453178010502556, 0, 1.21132753789727453178010502556, 2.25060759300894240501711562559, 3.18903907366865131288320582938, 4.24222845533843803943618371786, 5.06870680766319278913015395048, 5.60418548234518930767719640750, 5.94110041103914456544383732185, 7.24319797421775232631903345366, 7.61946981076169765809075698079

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