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Properties

Label 2-6480-1.1-c1-0-79
Degree $2$
Conductor $6480$
Sign $-1$
Analytic cond. $51.7430$
Root an. cond. $7.19326$
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 + 5·11-s + 3·17-s − 5·19-s − 6·23-s + 25-s − 10·29-s + 2·31-s + 4·37-s − 3·41-s − 3·43-s − 4·47-s − 7·49-s − 6·53-s − 5·55-s + 3·59-s + 2·61-s + 11·67-s + 14·71-s − 15·73-s − 10·79-s + 12·83-s − 3·85-s + 14·89-s + 5·95-s − 13·97-s − 12·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.727·17-s − 1.14·19-s − 1.25·23-s + 1/5·25-s − 1.85·29-s + 0.359·31-s + 0.657·37-s − 0.468·41-s − 0.457·43-s − 0.583·47-s − 49-s − 0.824·53-s − 0.674·55-s + 0.390·59-s + 0.256·61-s + 1.34·67-s + 1.66·71-s − 1.75·73-s − 1.12·79-s + 1.31·83-s − 0.325·85-s + 1.48·89-s + 0.512·95-s − 1.31·97-s − 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6480\)    =    \(2^{4} \cdot 3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(51.7430\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6480,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−7.81212256184125416581208696729, −6.84030052849159377245471985233, −6.36315767456006162547057716195, −5.62281187732482829008822990279, −4.64605279824741366722167485179, −3.89307308908582402198992282959, −3.47412284263825399275488195861, −2.17888958965834404694640156988, −1.35261254883323916209184071170, 0, 1.35261254883323916209184071170, 2.17888958965834404694640156988, 3.47412284263825399275488195861, 3.89307308908582402198992282959, 4.64605279824741366722167485179, 5.62281187732482829008822990279, 6.36315767456006162547057716195, 6.84030052849159377245471985233, 7.81212256184125416581208696729

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