Properties

Label 2-8001-1.1-c1-0-194
Degree $2$
Conductor $8001$
Sign $-1$
Analytic cond. $63.8883$
Root an. cond. $7.99301$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.55·2-s + 0.429·4-s + 1.78·5-s − 7-s + 2.44·8-s − 2.77·10-s − 0.515·11-s − 5.03·13-s + 1.55·14-s − 4.67·16-s + 0.0785·17-s + 4.96·19-s + 0.763·20-s + 0.803·22-s − 1.02·23-s − 1.83·25-s + 7.84·26-s − 0.429·28-s − 4.62·29-s + 2.09·31-s + 2.38·32-s − 0.122·34-s − 1.78·35-s + 5.01·37-s − 7.73·38-s + 4.35·40-s + 2.22·41-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.214·4-s + 0.796·5-s − 0.377·7-s + 0.865·8-s − 0.877·10-s − 0.155·11-s − 1.39·13-s + 0.416·14-s − 1.16·16-s + 0.0190·17-s + 1.13·19-s + 0.170·20-s + 0.171·22-s − 0.212·23-s − 0.366·25-s + 1.53·26-s − 0.0810·28-s − 0.858·29-s + 0.376·31-s + 0.422·32-s − 0.0210·34-s − 0.300·35-s + 0.824·37-s − 1.25·38-s + 0.689·40-s + 0.346·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
Sign: $-1$
Analytic conductor: \(63.8883\)
Root analytic conductor: \(7.99301\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
127 \( 1 - T \)
good2 \( 1 + 1.55T + 2T^{2} \)
5 \( 1 - 1.78T + 5T^{2} \)
11 \( 1 + 0.515T + 11T^{2} \)
13 \( 1 + 5.03T + 13T^{2} \)
17 \( 1 - 0.0785T + 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
23 \( 1 + 1.02T + 23T^{2} \)
29 \( 1 + 4.62T + 29T^{2} \)
31 \( 1 - 2.09T + 31T^{2} \)
37 \( 1 - 5.01T + 37T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 2.98T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 3.21T + 61T^{2} \)
67 \( 1 - 6.73T + 67T^{2} \)
71 \( 1 + 7.69T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 + 3.98T + 79T^{2} \)
83 \( 1 - 4.54T + 83T^{2} \)
89 \( 1 + 6.80T + 89T^{2} \)
97 \( 1 - 6.33T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.51082523327725661096800631848, −7.14969579231394748349786233721, −6.15845959533143997215648127230, −5.46091897735520380730094915151, −4.78540529926063898543154602355, −3.91217751865859907352404594328, −2.75406094859603973548801743570, −2.09777116581472295400036952388, −1.09405676251888750178416545756, 0, 1.09405676251888750178416545756, 2.09777116581472295400036952388, 2.75406094859603973548801743570, 3.91217751865859907352404594328, 4.78540529926063898543154602355, 5.46091897735520380730094915151, 6.15845959533143997215648127230, 7.14969579231394748349786233721, 7.51082523327725661096800631848

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