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  − 1.03·3-s − 2.90·5-s − 0.294·7-s − 1.93·9-s + 3.78·11-s − 3.15·13-s + 2.99·15-s − 6.61·17-s + 1.57·19-s + 0.304·21-s + 5.53·23-s + 3.42·25-s + 5.09·27-s + 0.673·29-s + 6.77·31-s − 3.91·33-s + 0.856·35-s + 2.01·37-s + 3.26·39-s + 4.28·41-s + 0.897·43-s + 5.61·45-s + 6.08·47-s − 6.91·49-s + 6.82·51-s + 4.70·53-s − 10.9·55-s + ⋯
L(s)  = 1  − 0.596·3-s − 1.29·5-s − 0.111·7-s − 0.644·9-s + 1.14·11-s − 0.876·13-s + 0.774·15-s − 1.60·17-s + 0.361·19-s + 0.0664·21-s + 1.15·23-s + 0.685·25-s + 0.980·27-s + 0.125·29-s + 1.21·31-s − 0.680·33-s + 0.144·35-s + 0.330·37-s + 0.522·39-s + 0.669·41-s + 0.136·43-s + 0.836·45-s + 0.886·47-s − 0.987·49-s + 0.956·51-s + 0.646·53-s − 1.48·55-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 \( 1 + 1.03T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 + 0.294T + 7T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 - 1.57T + 19T^{2} \)
23 \( 1 - 5.53T + 23T^{2} \)
29 \( 1 - 0.673T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 2.01T + 37T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 - 0.897T + 43T^{2} \)
47 \( 1 - 6.08T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 0.579T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 8.03T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 3.10T + 97T^{2} \)
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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

−7.65134789470005655093708094773, −6.91318781044171906274872079243, −6.49552801916525089598609478947, −5.57341553615742133963010070652, −4.59271964735781113600764068989, −4.30586522395089232600709049201, −3.25249300072962967340661415062, −2.48367581825186396224905107812, −0.977135235451922692975186942985, 0, 0.977135235451922692975186942985, 2.48367581825186396224905107812, 3.25249300072962967340661415062, 4.30586522395089232600709049201, 4.59271964735781113600764068989, 5.57341553615742133963010070652, 6.49552801916525089598609478947, 6.91318781044171906274872079243, 7.65134789470005655093708094773

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