Properties

Label 2-342-1.1-c1-0-6
Degree $2$
Conductor $342$
Sign $-1$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·7-s − 8-s − 4·13-s + 4·14-s + 16-s − 6·17-s + 19-s + 6·23-s − 5·25-s + 4·26-s − 4·28-s − 6·29-s + 2·31-s − 32-s + 6·34-s − 4·37-s − 38-s − 6·41-s − 4·43-s − 6·46-s − 6·47-s + 9·49-s + 5·50-s − 4·52-s − 6·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s + 1.25·23-s − 25-s + 0.784·26-s − 0.755·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s − 0.657·37-s − 0.162·38-s − 0.937·41-s − 0.609·43-s − 0.884·46-s − 0.875·47-s + 9/7·49-s + 0.707·50-s − 0.554·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{342} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−10.95189135954596859968721862656, −9.814927382770120407835166865620, −9.465004478872581430365849790992, −8.381778434403041270610948676969, −7.06756557503064388001592546481, −6.59547283064248335138694127148, −5.21083603966480863402338802043, −3.58677867797817326821260752741, −2.34092707995523044468378525917, 0, 2.34092707995523044468378525917, 3.58677867797817326821260752741, 5.21083603966480863402338802043, 6.59547283064248335138694127148, 7.06756557503064388001592546481, 8.381778434403041270610948676969, 9.465004478872581430365849790992, 9.814927382770120407835166865620, 10.95189135954596859968721862656

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