Properties

Label 2-6080-1.1-c1-0-118
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 3·7-s − 2·9-s − 5·13-s − 15-s − 3·17-s + 19-s + 3·21-s + 7·23-s + 25-s − 5·27-s + 29-s + 2·31-s − 3·35-s − 2·37-s − 5·39-s − 10·41-s + 6·43-s + 2·45-s + 8·47-s + 2·49-s − 3·51-s − 9·53-s + 57-s − 5·59-s − 4·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 1.38·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.654·21-s + 1.45·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s + 0.359·31-s − 0.507·35-s − 0.328·37-s − 0.800·39-s − 1.56·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.132·57-s − 0.650·59-s − 0.512·61-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: \(1\)
Selberg data: \((2,\ 6080,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−7.64754828690738031878730277844, −7.32103674047455755292357769219, −6.40866698888712532126911121020, −5.27647909838592918249759305503, −4.88809774471546246399252783679, −4.10622921369590355664734711516, −3.02150681403980172134722054388, −2.48468059649855058783767421873, −1.43949216133368788524626274844, 0, 1.43949216133368788524626274844, 2.48468059649855058783767421873, 3.02150681403980172134722054388, 4.10622921369590355664734711516, 4.88809774471546246399252783679, 5.27647909838592918249759305503, 6.40866698888712532126911121020, 7.32103674047455755292357769219, 7.64754828690738031878730277844

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