Properties

Label 2-1441-1.1-c1-0-15
Degree $2$
Conductor $1441$
Sign $1$
Analytic cond. $11.5064$
Root an. cond. $3.39211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.413·2-s − 2.92·3-s − 1.82·4-s − 1.23·5-s + 1.20·6-s + 5.16·7-s + 1.58·8-s + 5.54·9-s + 0.509·10-s − 11-s + 5.34·12-s − 5.31·13-s − 2.13·14-s + 3.60·15-s + 3.00·16-s + 5.77·17-s − 2.28·18-s − 5.78·19-s + 2.25·20-s − 15.1·21-s + 0.413·22-s − 0.288·23-s − 4.62·24-s − 3.48·25-s + 2.19·26-s − 7.42·27-s − 9.45·28-s + ⋯
L(s)  = 1  − 0.292·2-s − 1.68·3-s − 0.914·4-s − 0.551·5-s + 0.493·6-s + 1.95·7-s + 0.559·8-s + 1.84·9-s + 0.161·10-s − 0.301·11-s + 1.54·12-s − 1.47·13-s − 0.570·14-s + 0.930·15-s + 0.751·16-s + 1.40·17-s − 0.539·18-s − 1.32·19-s + 0.504·20-s − 3.29·21-s + 0.0880·22-s − 0.0600·23-s − 0.943·24-s − 0.696·25-s + 0.430·26-s − 1.42·27-s − 1.78·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $1$
Analytic conductor: \(11.5064\)
Root analytic conductor: \(3.39211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5137420231\)
\(L(\frac12)\) \(\approx\) \(0.5137420231\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
131 \( 1 - T \)
good2 \( 1 + 0.413T + 2T^{2} \)
3 \( 1 + 2.92T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 5.16T + 7T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 + 5.78T + 19T^{2} \)
23 \( 1 + 0.288T + 23T^{2} \)
29 \( 1 + 4.95T + 29T^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 - 8.22T + 37T^{2} \)
41 \( 1 + 3.54T + 41T^{2} \)
43 \( 1 + 1.58T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 9.49T + 53T^{2} \)
59 \( 1 - 2.41T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 9.78T + 71T^{2} \)
73 \( 1 + 9.62T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 - 4.18T + 83T^{2} \)
89 \( 1 - 1.89T + 89T^{2} \)
97 \( 1 - 2.15T + 97T^{2} \)
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

−9.870905833099336661663850779252, −8.556289943225714368024428507467, −7.76234780802403299471102934555, −7.42765904536875631333744922607, −5.97505045029619953747167741437, −5.11403149323952740088280798216, −4.79796758191947577301271138596, −4.02217541979774487878168969098, −1.85038180196971600640452305775, −0.60129285908409892929461275624, 0.60129285908409892929461275624, 1.85038180196971600640452305775, 4.02217541979774487878168969098, 4.79796758191947577301271138596, 5.11403149323952740088280798216, 5.97505045029619953747167741437, 7.42765904536875631333744922607, 7.76234780802403299471102934555, 8.556289943225714368024428507467, 9.870905833099336661663850779252

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