↑

Properties

Label 2-135240-1.1-c1-0-20
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s + 4·19-s − 23-s + 25-s − 27-s + 2·29-s + 4·33-s + 6·37-s + 2·39-s + 6·41-s − 8·43-s − 45-s − 4·47-s − 2·51-s + 2·53-s + 4·55-s − 4·57-s − 12·59-s + 2·61-s + 2·65-s + 8·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s − 0.280·51-s + 0.274·53-s + 0.539·55-s − 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.415209498\)
\(L(\frac12)\) \(\approx\) \(1.415209498\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−13.30667985843168, −12.89178910239255, −12.47829803785703, −12.00434507749839, −11.46209321106215, −11.20492593254400, −10.47449373971673, −10.18541816941566, −9.650225263405019, −9.225456859938610, −8.373270529252329, −8.025425555614108, −7.393283993966440, −7.323640208383977, −6.351592005635874, −6.042377371870633, −5.322588367780033, −4.874367105441485, −4.606909762860433, −3.644620658786657, −3.255729744783641, −2.542688180664379, −1.955636368807826, −0.9814555427284423, −0.4422393323682231, 0.4422393323682231, 0.9814555427284423, 1.955636368807826, 2.542688180664379, 3.255729744783641, 3.644620658786657, 4.606909762860433, 4.874367105441485, 5.322588367780033, 6.042377371870633, 6.351592005635874, 7.323640208383977, 7.393283993966440, 8.025425555614108, 8.373270529252329, 9.225456859938610, 9.650225263405019, 10.18541816941566, 10.47449373971673, 11.20492593254400, 11.46209321106215, 12.00434507749839, 12.47829803785703, 12.89178910239255, 13.30667985843168

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