↑

Properties

Label 2-510e2-1.1-c1-0-70
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 + 7·13-s − 7·19-s + 11·31-s − 11·37-s + 13·43-s + 18·49-s − 13·61-s − 5·67-s + 10·73-s − 4·79-s − 35·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s − 1.60·19-s + 1.97·31-s − 1.80·37-s + 1.98·43-s + 18/7·49-s − 1.66·61-s − 0.610·67-s + 1.17·73-s − 0.450·79-s − 3.66·91-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 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
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

−12.95988535648760, −12.70983016014675, −12.18187709095260, −11.83390623400453, −10.96821447603288, −10.70860362984399, −10.37548954244851, −9.839925324220572, −9.262451603657389, −8.870745168113005, −8.515896966204836, −8.032471646046643, −7.259936150237357, −6.770569538787454, −6.308791489694083, −6.092170888079138, −5.699516447010922, −4.776130702764907, −4.187281481995851, −3.789302355034128, −3.291082398456513, −2.793623334309341, −2.196213363556282, −1.362912861209287, −0.7129457938307863, 0, 0.7129457938307863, 1.362912861209287, 2.196213363556282, 2.793623334309341, 3.291082398456513, 3.789302355034128, 4.187281481995851, 4.776130702764907, 5.699516447010922, 6.092170888079138, 6.308791489694083, 6.770569538787454, 7.259936150237357, 8.032471646046643, 8.515896966204836, 8.870745168113005, 9.262451603657389, 9.839925324220572, 10.37548954244851, 10.70860362984399, 10.96821447603288, 11.83390623400453, 12.18187709095260, 12.70983016014675, 12.95988535648760

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