Properties

Label 2-135-1.1-c3-0-7
Degree $2$
Conductor $135$
Sign $-1$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.45·2-s + 21.7·4-s − 5·5-s − 11.8·7-s − 75.3·8-s + 27.2·10-s + 56.2·11-s + 34.5·13-s + 64.4·14-s + 236.·16-s − 39.2·17-s − 146.·19-s − 108.·20-s − 306.·22-s + 23.5·23-s + 25·25-s − 188.·26-s − 257.·28-s − 161.·29-s − 29.5·31-s − 689.·32-s + 214.·34-s + 59.0·35-s − 217.·37-s + 800.·38-s + 376.·40-s − 142.·41-s + ⋯
L(s)  = 1  − 1.92·2-s + 2.72·4-s − 0.447·5-s − 0.637·7-s − 3.32·8-s + 0.863·10-s + 1.54·11-s + 0.738·13-s + 1.23·14-s + 3.69·16-s − 0.560·17-s − 1.76·19-s − 1.21·20-s − 2.97·22-s + 0.213·23-s + 0.200·25-s − 1.42·26-s − 1.73·28-s − 1.03·29-s − 0.171·31-s − 3.81·32-s + 1.08·34-s + 0.285·35-s − 0.967·37-s + 3.41·38-s + 1.48·40-s − 0.541·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + 5T \)
good2 \( 1 + 5.45T + 8T^{2} \)
7 \( 1 + 11.8T + 343T^{2} \)
11 \( 1 - 56.2T + 1.33e3T^{2} \)
13 \( 1 - 34.5T + 2.19e3T^{2} \)
17 \( 1 + 39.2T + 4.91e3T^{2} \)
19 \( 1 + 146.T + 6.85e3T^{2} \)
23 \( 1 - 23.5T + 1.21e4T^{2} \)
29 \( 1 + 161.T + 2.43e4T^{2} \)
31 \( 1 + 29.5T + 2.97e4T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 + 142.T + 6.89e4T^{2} \)
43 \( 1 + 468.T + 7.95e4T^{2} \)
47 \( 1 + 394.T + 1.03e5T^{2} \)
53 \( 1 - 134.T + 1.48e5T^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 - 259.T + 2.26e5T^{2} \)
67 \( 1 - 445.T + 3.00e5T^{2} \)
71 \( 1 + 560.T + 3.57e5T^{2} \)
73 \( 1 + 88.6T + 3.89e5T^{2} \)
79 \( 1 - 450.T + 4.93e5T^{2} \)
83 \( 1 + 284.T + 5.71e5T^{2} \)
89 \( 1 + 625.T + 7.04e5T^{2} \)
97 \( 1 + 193.T + 9.12e5T^{2} \)
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

−11.72153793629543734917912107376, −11.04168591207642088134434207604, −9.977270890274071246218878528327, −8.948622735300731405871099107539, −8.393860479009441564492549028495, −6.89033249329496026564965502695, −6.37366784945241391825937438286, −3.59870478363771943305380763226, −1.71328184353382354875338618041, 0, 1.71328184353382354875338618041, 3.59870478363771943305380763226, 6.37366784945241391825937438286, 6.89033249329496026564965502695, 8.393860479009441564492549028495, 8.948622735300731405871099107539, 9.977270890274071246218878528327, 11.04168591207642088134434207604, 11.72153793629543734917912107376

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