Properties

Label 2-1260-1.1-c1-0-2
Degree $2$
Conductor $1260$
Sign $1$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 2·11-s + 4·13-s − 2·17-s + 2·19-s − 4·23-s + 25-s − 6·29-s − 2·31-s − 35-s + 10·37-s + 10·41-s + 12·43-s + 8·47-s + 49-s − 2·55-s + 8·59-s − 2·61-s − 4·65-s − 12·67-s + 10·71-s + 4·73-s + 2·77-s + 12·83-s + 2·85-s − 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 1.56·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.269·55-s + 1.04·59-s − 0.256·61-s − 0.496·65-s − 1.46·67-s + 1.18·71-s + 0.468·73-s + 0.227·77-s + 1.31·83-s + 0.216·85-s − 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695840180\)
\(L(\frac12)\) \(\approx\) \(1.695840180\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−9.459268171549549227560669133895, −8.977489293175671900227133863971, −7.972522006250822658500010164715, −7.40282489558306515108943317960, −6.27575535344744739082506805421, −5.62102206398535181378106731263, −4.28120249162261891993830881046, −3.79614462524231983446903246185, −2.38904886593758690492350473788, −1.01812237326846089124898046997, 1.01812237326846089124898046997, 2.38904886593758690492350473788, 3.79614462524231983446903246185, 4.28120249162261891993830881046, 5.62102206398535181378106731263, 6.27575535344744739082506805421, 7.40282489558306515108943317960, 7.972522006250822658500010164715, 8.977489293175671900227133863971, 9.459268171549549227560669133895

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