Properties

Label 1-2541-2541.89-r0-0-0
Degree $1$
Conductor $2541$
Sign $-0.715 - 0.698i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.981 + 0.189i)5-s + (0.654 + 0.755i)8-s + (−0.0475 − 0.998i)10-s + (−0.841 − 0.540i)13-s + (0.580 − 0.814i)16-s + (0.928 + 0.371i)17-s + (−0.928 + 0.371i)19-s + (−0.959 + 0.281i)20-s + (−0.580 + 0.814i)23-s + (0.928 + 0.371i)25-s + (−0.327 + 0.945i)26-s + (0.142 − 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.928 − 0.371i)32-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.981 + 0.189i)5-s + (0.654 + 0.755i)8-s + (−0.0475 − 0.998i)10-s + (−0.841 − 0.540i)13-s + (0.580 − 0.814i)16-s + (0.928 + 0.371i)17-s + (−0.928 + 0.371i)19-s + (−0.959 + 0.281i)20-s + (−0.580 + 0.814i)23-s + (0.928 + 0.371i)25-s + (−0.327 + 0.945i)26-s + (0.142 − 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.928 − 0.371i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ -0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4280631257 - 1.051349086i\)
\(L(\frac12)\) \(\approx\) \(0.4280631257 - 1.051349086i\)
\(L(1)\) \(\approx\) \(0.8133425931 - 0.4470318467i\)
\(L(1)\) \(\approx\) \(0.8133425931 - 0.4470318467i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
5 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (0.928 + 0.371i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
23 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.888 - 0.458i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.723 + 0.690i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.995 + 0.0950i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.55022206294077819155199664762, −18.68331892655481640336262866834, −18.19855230095774323186199626839, −17.40593109262016240960261807449, −16.76709927738586163620833233001, −16.43147270592348745388851201274, −15.42303798820942504678639654582, −14.57204000866113557431147399169, −14.19196399067858404774685979596, −13.485137346438233062217615914108, −12.6324724499684696463420741355, −12.028552310017095774698919735068, −10.57701594462626883907612586280, −10.18255090279765282175816696478, −9.369421494527139630649429317866, −8.75131875801474101017775163316, −8.04063036986321077027079072047, −6.87483497868842676829422933448, −6.6671869442116614344614416806, −5.57979681670016938176080558306, −5.028245311391910468517846917805, −4.31068871226460106115875756227, −3.057346315369422402241510809245, −1.97975022994983751329686044035, −1.0761275659101204073509503969, 0.41520427550030721934055166210, 1.771613591130189951021742388, 2.140524549799158516229573634083, 3.18377762667907944663186982364, 3.89390029710747640027151088233, 5.011791232959664265602976839687, 5.5977895171254304931371687345, 6.516858461707225711508907263237, 7.69426906434759569462996003160, 8.2152698786199145544087195153, 9.29428035029633213449814227059, 9.8351588093789710315208474487, 10.35509401241261567830566740917, 11.05818330852672156150791281227, 12.09962695170346082561434590642, 12.52568119724004546982697361018, 13.373707624698168206781650789549, 13.96435731844405813497231555793, 14.66683016324329971513447661093, 15.47443544126537621046815416430, 16.8481750531044903841971362443, 17.1233055628624550655227347456, 17.73133824439336769010397066205, 18.61803132736431969088920863498, 19.09382724521091672301478435513

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