Properties

Label 2-3311-3311.3310-c0-0-11
Degree $2$
Conductor $3311$
Sign $1$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $1$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3311} (3310, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4789155944\)
\(L(\frac12)\) \(\approx\) \(0.4789155944\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
13 \( ( 1 - T )^{2} \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T + T^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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

−8.862262329947913428343529070066, −8.258538725477221090726449746241, −7.43927517609107944454842365043, −6.72363180263083440484828215114, −5.90167615767299762479450629027, −4.90290354737603484033044377537, −4.29092155912826760244323363959, −3.47490674582878888548143372136, −1.63069526623862256761644080483, −0.819001488182938398376713417081, 0.819001488182938398376713417081, 1.63069526623862256761644080483, 3.47490674582878888548143372136, 4.29092155912826760244323363959, 4.90290354737603484033044377537, 5.90167615767299762479450629027, 6.72363180263083440484828215114, 7.43927517609107944454842365043, 8.258538725477221090726449746241, 8.862262329947913428343529070066

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