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  − 3.30·3-s + 0.148·5-s − 4.48·7-s + 7.92·9-s − 5.76·11-s + 2.52·13-s − 0.490·15-s − 1.39·17-s + 6.53·19-s + 14.8·21-s − 1.51·23-s − 4.97·25-s − 16.2·27-s − 7.16·29-s + 4.58·31-s + 19.0·33-s − 0.665·35-s − 3.08·37-s − 8.35·39-s + 1.44·41-s − 0.693·43-s + 1.17·45-s + 0.309·47-s + 13.1·49-s + 4.61·51-s + 9.64·53-s − 0.856·55-s + ⋯
L(s)  = 1  − 1.90·3-s + 0.0663·5-s − 1.69·7-s + 2.64·9-s − 1.73·11-s + 0.701·13-s − 0.126·15-s − 0.338·17-s + 1.49·19-s + 3.23·21-s − 0.315·23-s − 0.995·25-s − 3.13·27-s − 1.32·29-s + 0.823·31-s + 3.31·33-s − 0.112·35-s − 0.506·37-s − 1.33·39-s + 0.225·41-s − 0.105·43-s + 0.175·45-s + 0.0451·47-s + 1.87·49-s + 0.646·51-s + 1.32·53-s − 0.115·55-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 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 - 0.148T + 5T^{2} \)
7 \( 1 + 4.48T + 7T^{2} \)
11 \( 1 + 5.76T + 11T^{2} \)
13 \( 1 - 2.52T + 13T^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 - 6.53T + 19T^{2} \)
23 \( 1 + 1.51T + 23T^{2} \)
29 \( 1 + 7.16T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 - 1.44T + 41T^{2} \)
43 \( 1 + 0.693T + 43T^{2} \)
47 \( 1 - 0.309T + 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 - 14.8T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 7.49T + 71T^{2} \)
73 \( 1 - 1.54T + 73T^{2} \)
79 \( 1 + 9.02T + 79T^{2} \)
83 \( 1 + 0.305T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.28570015763328329327691218208, −7.00905921879213438052263735962, −6.06438605927390559719770354928, −5.64134031055791415716367042913, −5.23824903777558962655245303121, −4.11319203656644440930934599866, −3.39816074476015142390810633255, −2.25635320155073654030778338295, −0.814987989596458639881007945165, 0, 0.814987989596458639881007945165, 2.25635320155073654030778338295, 3.39816074476015142390810633255, 4.11319203656644440930934599866, 5.23824903777558962655245303121, 5.64134031055791415716367042913, 6.06438605927390559719770354928, 7.00905921879213438052263735962, 7.28570015763328329327691218208

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