Properties

Label 2-15680-1.1-c1-0-66
Degree $2$
Conductor $15680$
Sign $-1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 9-s + 3·11-s − 5·13-s − 2·15-s + 6·17-s − 19-s − 3·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s − 6·33-s − 11·37-s + 10·39-s + 3·41-s − 10·43-s + 45-s − 3·47-s − 12·51-s − 3·53-s + 3·55-s + 2·57-s + 4·61-s − 5·65-s − 4·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.38·13-s − 0.516·15-s + 1.45·17-s − 0.229·19-s − 0.625·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.04·33-s − 1.80·37-s + 1.60·39-s + 0.468·41-s − 1.52·43-s + 0.149·45-s − 0.437·47-s − 1.68·51-s − 0.412·53-s + 0.404·55-s + 0.264·57-s + 0.512·61-s − 0.620·65-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.34755569981787, −15.99852193001746, −15.07849731160264, −14.51767287688415, −14.17121810958494, −13.56873195431973, −12.63157966239224, −12.20146268896425, −11.88480101959004, −11.42943581962299, −10.43266977821791, −10.14801942173579, −9.735721051352669, −8.843610733299129, −8.274679932556058, −7.414013167148964, −6.854547010441639, −6.223299008979212, −5.770909219907725, −4.927060831793643, −4.731750005131754, −3.593517906534936, −2.871979854685591, −1.856951675032093, −1.029387530647764, 0, 1.029387530647764, 1.856951675032093, 2.871979854685591, 3.593517906534936, 4.731750005131754, 4.927060831793643, 5.770909219907725, 6.223299008979212, 6.854547010441639, 7.414013167148964, 8.274679932556058, 8.843610733299129, 9.735721051352669, 10.14801942173579, 10.43266977821791, 11.42943581962299, 11.88480101959004, 12.20146268896425, 12.63157966239224, 13.56873195431973, 14.17121810958494, 14.51767287688415, 15.07849731160264, 15.99852193001746, 16.34755569981787

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