Properties

Label 2-6080-1.1-c1-0-15
Degree $2$
Conductor $6080$
Sign $1$
Analytic cond. $48.5490$
Root an. cond. $6.96771$
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  − 5-s − 3·9-s − 4·11-s + 6·13-s − 6·17-s − 19-s − 8·23-s + 25-s + 2·29-s − 2·37-s + 2·41-s + 4·43-s + 3·45-s + 8·47-s − 7·49-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s − 6·65-s + 8·67-s − 8·71-s + 2·73-s + 8·79-s + 9·81-s + 4·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s − 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.229·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.447·45-s + 1.16·47-s − 49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.744·65-s + 0.977·67-s − 0.949·71-s + 0.234·73-s + 0.900·79-s + 81-s + 0.439·83-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6080\)    =    \(2^{6} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(48.5490\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.006238935\)
\(L(\frac12)\) \(\approx\) \(1.006238935\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.202970558214809706820905364823, −7.52473765533362103025714467932, −6.47423013670024813740129837470, −6.02039957027640883531787418645, −5.27993492779365324438642971585, −4.32846397493408277296634806263, −3.69279128054701904393939711757, −2.75423773404278858496955161486, −1.99466167783360732353573227592, −0.50004200889958251484664781302, 0.50004200889958251484664781302, 1.99466167783360732353573227592, 2.75423773404278858496955161486, 3.69279128054701904393939711757, 4.32846397493408277296634806263, 5.27993492779365324438642971585, 6.02039957027640883531787418645, 6.47423013670024813740129837470, 7.52473765533362103025714467932, 8.202970558214809706820905364823

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