Properties

Label 2-51e2-1.1-c1-0-80
Degree $2$
Conductor $2601$
Sign $-1$
Analytic cond. $20.7690$
Root an. cond. $4.55731$
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  − 2·2-s + 2·4-s + 5-s + 2·7-s − 2·10-s + 3·11-s − 5·13-s − 4·14-s − 4·16-s − 19-s + 2·20-s − 6·22-s + 7·23-s − 4·25-s + 10·26-s + 4·28-s − 6·29-s − 4·31-s + 8·32-s + 2·35-s − 10·37-s + 2·38-s − 9·41-s + 43-s + 6·44-s − 14·46-s − 12·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s + 0.755·7-s − 0.632·10-s + 0.904·11-s − 1.38·13-s − 1.06·14-s − 16-s − 0.229·19-s + 0.447·20-s − 1.27·22-s + 1.45·23-s − 4/5·25-s + 1.96·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 1.41·32-s + 0.338·35-s − 1.64·37-s + 0.324·38-s − 1.40·41-s + 0.152·43-s + 0.904·44-s − 2.06·46-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(20.7690\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−8.558687788277825826647659579001, −7.916377415932892056898606968542, −7.10999490724463205156992109664, −6.63286098408952987088726943518, −5.29144791172147551548203273912, −4.73696182344134369376729942173, −3.45923135563791354670887160785, −2.05003307500449234278531577643, −1.52489258848714603875565670450, 0, 1.52489258848714603875565670450, 2.05003307500449234278531577643, 3.45923135563791354670887160785, 4.73696182344134369376729942173, 5.29144791172147551548203273912, 6.63286098408952987088726943518, 7.10999490724463205156992109664, 7.916377415932892056898606968542, 8.558687788277825826647659579001

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