Properties

Label 2-141-1.1-c1-0-5
Degree $2$
Conductor $141$
Sign $1$
Analytic cond. $1.12589$
Root an. cond. $1.06107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 3·7-s + 9-s − 2·10-s + 11-s + 2·12-s − 2·13-s − 6·14-s − 15-s − 4·16-s + 2·17-s + 2·18-s + 6·19-s − 2·20-s − 3·21-s + 2·22-s + 3·23-s − 4·25-s − 4·26-s + 27-s − 6·28-s + 3·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s − 1.60·14-s − 0.258·15-s − 16-s + 0.485·17-s + 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.654·21-s + 0.426·22-s + 0.625·23-s − 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.13·28-s + 0.557·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141\)    =    \(3 \cdot 47\)
Sign: $1$
Analytic conductor: \(1.12589\)
Root analytic conductor: \(1.06107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{141} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 141,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.107535578\)
\(L(\frac12)\) \(\approx\) \(2.107535578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
47 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−13.27824273167304406003722016011, −12.33792187853331891937545023201, −11.68018802766311299547378241988, −10.03012465082032508207642749183, −9.085897389420589721942660476599, −7.52539857035144694302700552711, −6.46487870715497229983731599323, −5.14885571821443324588162204875, −3.77257881491072191084851286374, −2.93657054326743540839942565323, 2.93657054326743540839942565323, 3.77257881491072191084851286374, 5.14885571821443324588162204875, 6.46487870715497229983731599323, 7.52539857035144694302700552711, 9.085897389420589721942660476599, 10.03012465082032508207642749183, 11.68018802766311299547378241988, 12.33792187853331891937545023201, 13.27824273167304406003722016011

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