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Properties

Label 2-10008-1.1-c1-0-5
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 5·11-s − 3·13-s − 2·17-s − 6·19-s + 2·23-s + 4·25-s + 3·29-s + 3·31-s − 3·35-s + 2·37-s + 6·41-s − 4·43-s − 6·49-s − 15·55-s + 14·59-s + 12·61-s + 9·65-s − 13·67-s − 15·71-s − 14·73-s + 5·77-s + 79-s + 7·83-s + 6·85-s + 9·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 1.50·11-s − 0.832·13-s − 0.485·17-s − 1.37·19-s + 0.417·23-s + 4/5·25-s + 0.557·29-s + 0.538·31-s − 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 6/7·49-s − 2.02·55-s + 1.82·59-s + 1.53·61-s + 1.11·65-s − 1.58·67-s − 1.78·71-s − 1.63·73-s + 0.569·77-s + 0.112·79-s + 0.768·83-s + 0.650·85-s + 0.953·89-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: \(1\)
Selberg data: \((2,\ 10008,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
139 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.91613869000095, −16.27870403832286, −15.86384997212788, −14.95934759414407, −14.74243806717491, −14.41839486467068, −13.34167314402057, −12.86878924227218, −12.00019317004089, −11.80344002947073, −11.29008107658468, −10.57364830647907, −9.909924586969989, −9.013284199559351, −8.641922160982934, −7.981771697022123, −7.310914352884814, −6.719326018193500, −6.164981867964294, −5.031065575133643, −4.284653589819178, −4.098630295370669, −3.100477122272055, −2.177611147775203, −1.101298135226797, 0, 1.101298135226797, 2.177611147775203, 3.100477122272055, 4.098630295370669, 4.284653589819178, 5.031065575133643, 6.164981867964294, 6.719326018193500, 7.310914352884814, 7.981771697022123, 8.641922160982934, 9.013284199559351, 9.909924586969989, 10.57364830647907, 11.29008107658468, 11.80344002947073, 12.00019317004089, 12.86878924227218, 13.34167314402057, 14.41839486467068, 14.74243806717491, 14.95934759414407, 15.86384997212788, 16.27870403832286, 16.91613869000095

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