↑

Properties

Label 2-510e2-1.1-c1-0-61
Degree $2$
Conductor $260100$
Sign $-1$
Analytic cond. $2076.90$
Root an. cond. $45.5731$
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  − 5·7-s − 2·13-s + 8·19-s − 4·31-s + 37-s − 8·43-s + 18·49-s − 13·61-s − 5·67-s + 10·73-s − 4·79-s + 10·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.88·7-s − 0.554·13-s + 1.83·19-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s − 1.66·61-s − 0.610·67-s + 1.17·73-s − 0.450·79-s + 1.04·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2076.90\)
Root analytic conductor: \(45.5731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 260100,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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

−12.99774340123399, −12.64231335216605, −12.10113453009953, −11.84448119060290, −11.25040914173723, −10.62221783218351, −10.16173104370611, −9.757609747902387, −9.396456905720575, −9.099333559085175, −8.440490162927384, −7.757978735936350, −7.298697115279027, −6.995989496017276, −6.413776062503138, −5.940459439585251, −5.496941012950206, −4.935890907362993, −4.342074034330415, −3.521947832554272, −3.332262098302064, −2.857970473243585, −2.199082673247409, −1.401511768342529, −0.6388489234898537, 0, 0.6388489234898537, 1.401511768342529, 2.199082673247409, 2.857970473243585, 3.332262098302064, 3.521947832554272, 4.342074034330415, 4.935890907362993, 5.496941012950206, 5.940459439585251, 6.413776062503138, 6.995989496017276, 7.298697115279027, 7.757978735936350, 8.440490162927384, 9.099333559085175, 9.396456905720575, 9.757609747902387, 10.16173104370611, 10.62221783218351, 11.25040914173723, 11.84448119060290, 12.10113453009953, 12.64231335216605, 12.99774340123399

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