Properties

Label 2-10080-1.1-c1-0-34
Degree $2$
Conductor $10080$
Sign $-1$
Analytic cond. $80.4892$
Root an. cond. $8.97157$
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  − 5-s + 7-s − 3·11-s + 13-s + 17-s − 4·19-s + 4·23-s + 25-s + 9·29-s − 6·31-s − 35-s − 8·37-s − 6·41-s + 8·43-s − 7·47-s + 49-s + 8·53-s + 3·55-s + 4·59-s + 10·61-s − 65-s + 8·67-s − 12·71-s − 14·73-s − 3·77-s + 5·79-s − 16·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.904·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.67·29-s − 1.07·31-s − 0.169·35-s − 1.31·37-s − 0.937·41-s + 1.21·43-s − 1.02·47-s + 1/7·49-s + 1.09·53-s + 0.404·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s + 0.977·67-s − 1.42·71-s − 1.63·73-s − 0.341·77-s + 0.562·79-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10080\)    =    \(2^{5} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(80.4892\)
Root analytic conductor: \(8.97157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10080} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10080,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−16.99594045986195, −16.08323711980662, −15.93465613466272, −15.19296209594103, −14.64729464900630, −14.17914217383901, −13.28574255836248, −12.99633715875645, −12.24316522682511, −11.73010426267058, −11.01023763485074, −10.51095954577824, −10.07514877946792, −9.066710865019850, −8.457959788225414, −8.156588818474357, −7.183164344024145, −6.865355496568560, −5.854603186991252, −5.210640775106113, −4.596811876919944, −3.804914204851000, −3.008130688660900, −2.219900845813494, −1.171375200635309, 0, 1.171375200635309, 2.219900845813494, 3.008130688660900, 3.804914204851000, 4.596811876919944, 5.210640775106113, 5.854603186991252, 6.865355496568560, 7.183164344024145, 8.156588818474357, 8.457959788225414, 9.066710865019850, 10.07514877946792, 10.51095954577824, 11.01023763485074, 11.73010426267058, 12.24316522682511, 12.99633715875645, 13.28574255836248, 14.17914217383901, 14.64729464900630, 15.19296209594103, 15.93465613466272, 16.08323711980662, 16.99594045986195

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