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Properties

Label 2-10008-1.1-c1-0-0
Degree $2$
Conductor $10008$
Sign $1$
Analytic cond. $79.9142$
Root an. cond. $8.93947$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 11-s + 3·13-s + 5·17-s + 3·19-s + 4·23-s − 5·25-s − 6·31-s − 37-s − 43-s + 9·47-s − 3·49-s − 9·53-s + 4·59-s + 12·61-s + 2·67-s + 12·71-s − 2·73-s − 2·77-s + 10·79-s + 8·83-s + 12·89-s − 6·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.301·11-s + 0.832·13-s + 1.21·17-s + 0.688·19-s + 0.834·23-s − 25-s − 1.07·31-s − 0.164·37-s − 0.152·43-s + 1.31·47-s − 3/7·49-s − 1.23·53-s + 0.520·59-s + 1.53·61-s + 0.244·67-s + 1.42·71-s − 0.234·73-s − 0.227·77-s + 1.12·79-s + 0.878·83-s + 1.27·89-s − 0.628·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10008\)    =    \(2^{3} \cdot 3^{2} \cdot 139\)
Sign: $1$
Analytic conductor: \(79.9142\)
Root analytic conductor: \(8.93947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.058297177\)
\(L(\frac12)\) \(\approx\) \(2.058297177\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
139 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.39942974826106, −16.21305939469332, −15.60447422563583, −14.89530710242973, −14.35519124445322, −13.69138249839763, −13.25724652627525, −12.53972956054355, −12.10602620571574, −11.35164780067615, −10.84972891812002, −10.09607134225587, −9.478056392995125, −9.138796721620939, −8.204363342682387, −7.684681494794197, −6.911451566782573, −6.341605667974827, −5.575641850903179, −5.119493054414429, −3.819817602594139, −3.603225090158462, −2.714127066169684, −1.608433438840949, −0.7073728885283581, 0.7073728885283581, 1.608433438840949, 2.714127066169684, 3.603225090158462, 3.819817602594139, 5.119493054414429, 5.575641850903179, 6.341605667974827, 6.911451566782573, 7.684681494794197, 8.204363342682387, 9.138796721620939, 9.478056392995125, 10.09607134225587, 10.84972891812002, 11.35164780067615, 12.10602620571574, 12.53972956054355, 13.25724652627525, 13.69138249839763, 14.35519124445322, 14.89530710242973, 15.60447422563583, 16.21305939469332, 16.39942974826106

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