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  + 1.77·3-s − 3.34·5-s + 0.838·7-s + 0.146·9-s − 0.297·11-s − 1.29·13-s − 5.93·15-s + 2.69·17-s − 2.59·19-s + 1.48·21-s + 5.00·23-s + 6.17·25-s − 5.06·27-s + 4.26·29-s − 1.16·31-s − 0.527·33-s − 2.80·35-s + 6.31·37-s − 2.29·39-s + 0.0977·41-s + 7.74·43-s − 0.489·45-s − 4.23·47-s − 6.29·49-s + 4.77·51-s − 3.41·53-s + 0.993·55-s + ⋯
L(s)  = 1  + 1.02·3-s − 1.49·5-s + 0.316·7-s + 0.0488·9-s − 0.0896·11-s − 0.359·13-s − 1.53·15-s + 0.652·17-s − 0.594·19-s + 0.324·21-s + 1.04·23-s + 1.23·25-s − 0.974·27-s + 0.792·29-s − 0.208·31-s − 0.0917·33-s − 0.473·35-s + 1.03·37-s − 0.367·39-s + 0.0152·41-s + 1.18·43-s − 0.0729·45-s − 0.618·47-s − 0.899·49-s + 0.668·51-s − 0.469·53-s + 0.134·55-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 \( 1 - 1.77T + 3T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 - 0.838T + 7T^{2} \)
11 \( 1 + 0.297T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 - 5.00T + 23T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + 1.16T + 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 - 0.0977T + 41T^{2} \)
43 \( 1 - 7.74T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 + 3.41T + 53T^{2} \)
59 \( 1 - 5.04T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 4.02T + 67T^{2} \)
71 \( 1 + 2.77T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 8.20T + 97T^{2} \)
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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

−7.79403846207248460376938403071, −7.39224231077526376183990483553, −6.49658014713466981709336060352, −5.46067898165717495958259276480, −4.54029663756434044737871088216, −4.03034130277860698754929988465, −3.10016441441532391663132384525, −2.68977379715165831375602581287, −1.33983056017495970504438475064, 0, 1.33983056017495970504438475064, 2.68977379715165831375602581287, 3.10016441441532391663132384525, 4.03034130277860698754929988465, 4.54029663756434044737871088216, 5.46067898165717495958259276480, 6.49658014713466981709336060352, 7.39224231077526376183990483553, 7.79403846207248460376938403071

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