↑

Properties

Label 2-15e2-5.4-c3-0-11
Degree $2$
Conductor $225$
Sign $0.447 - 0.894i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5i·2-s − 17·4-s − 30i·7-s − 45i·8-s + 50·11-s + 20i·13-s + 150·14-s + 89·16-s − 10i·17-s + 44·19-s + 250i·22-s − 120i·23-s − 100·26-s + 510i·28-s + 50·29-s + ⋯
L(s)  = 1  + 1.76i·2-s − 2.12·4-s − 1.61i·7-s − 1.98i·8-s + 1.37·11-s + 0.426i·13-s + 2.86·14-s + 1.39·16-s − 0.142i·17-s + 0.531·19-s + 2.42i·22-s − 1.08i·23-s − 0.754·26-s + 3.44i·28-s + 0.320·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.32233 + 0.817247i\)
\(L(\frac12)\) \(\approx\) \(1.32233 + 0.817247i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 5iT - 8T^{2} \)
7 \( 1 + 30iT - 343T^{2} \)
11 \( 1 - 50T + 1.33e3T^{2} \)
13 \( 1 - 20iT - 2.19e3T^{2} \)
17 \( 1 + 10iT - 4.91e3T^{2} \)
19 \( 1 - 44T + 6.85e3T^{2} \)
23 \( 1 + 120iT - 1.21e4T^{2} \)
29 \( 1 - 50T + 2.43e4T^{2} \)
31 \( 1 - 108T + 2.97e4T^{2} \)
37 \( 1 + 40iT - 5.06e4T^{2} \)
41 \( 1 - 400T + 6.89e4T^{2} \)
43 \( 1 + 280iT - 7.95e4T^{2} \)
47 \( 1 + 280iT - 1.03e5T^{2} \)
53 \( 1 - 610iT - 1.48e5T^{2} \)
59 \( 1 + 50T + 2.05e5T^{2} \)
61 \( 1 + 518T + 2.26e5T^{2} \)
67 \( 1 + 180iT - 3.00e5T^{2} \)
71 \( 1 - 700T + 3.57e5T^{2} \)
73 \( 1 - 410iT - 3.89e5T^{2} \)
79 \( 1 - 516T + 4.93e5T^{2} \)
83 \( 1 + 660iT - 5.71e5T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + 1.63e3iT - 9.12e5T^{2} \)
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

−12.17164731416797132960391413615, −10.80871253320518527654999476492, −9.657245757134067183327530444274, −8.776486951253810735962028177705, −7.62866010655286319149762579498, −6.92123420797246701502859318480, −6.18243108001068588690407687760, −4.64671016246611190302920272824, −3.91793993002735070793711826197, −0.792100579264464299029250631769, 1.27265649963728689131021305035, 2.54338174539633863164390961164, 3.61787867834486799364124222991, 4.98422031162526763913619135692, 6.18986797736711358031040191399, 8.145144510953457086409545476407, 9.274144744209553996382140456511, 9.541088492914208136454206434538, 10.94299511453729755118206110675, 11.77435758496017140313736896269

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