Properties

Label 2-15e2-15.8-c3-0-16
Degree $2$
Conductor $225$
Sign $-0.999 - 0.0387i$
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  + (−1.07 − 1.07i)2-s − 5.70i·4-s + (7.85 − 7.85i)7-s + (−14.6 + 14.6i)8-s − 33.7i·11-s + (−25.7 − 25.7i)13-s − 16.8·14-s − 14.2·16-s + (66.3 + 66.3i)17-s − 128. i·19-s + (−36.0 + 36.0i)22-s + (−110. + 110. i)23-s + 55.0i·26-s + (−44.8 − 44.8i)28-s − 268.·29-s + ⋯
L(s)  = 1  + (−0.378 − 0.378i)2-s − 0.713i·4-s + (0.424 − 0.424i)7-s + (−0.648 + 0.648i)8-s − 0.923i·11-s + (−0.548 − 0.548i)13-s − 0.321·14-s − 0.222·16-s + (0.946 + 0.946i)17-s − 1.55i·19-s + (−0.349 + 0.349i)22-s + (−1.00 + 1.00i)23-s + 0.415i·26-s + (−0.302 − 0.302i)28-s − 1.72·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.999 - 0.0387i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.999 - 0.0387i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0153303 + 0.790662i\)
\(L(\frac12)\) \(\approx\) \(0.0153303 + 0.790662i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.07 + 1.07i)T + 8iT^{2} \)
7 \( 1 + (-7.85 + 7.85i)T - 343iT^{2} \)
11 \( 1 + 33.7iT - 1.33e3T^{2} \)
13 \( 1 + (25.7 + 25.7i)T + 2.19e3iT^{2} \)
17 \( 1 + (-66.3 - 66.3i)T + 4.91e3iT^{2} \)
19 \( 1 + 128. iT - 6.85e3T^{2} \)
23 \( 1 + (110. - 110. i)T - 1.21e4iT^{2} \)
29 \( 1 + 268.T + 2.43e4T^{2} \)
31 \( 1 + 2.91T + 2.97e4T^{2} \)
37 \( 1 + (115. - 115. i)T - 5.06e4iT^{2} \)
41 \( 1 + 251. iT - 6.89e4T^{2} \)
43 \( 1 + (340. + 340. i)T + 7.95e4iT^{2} \)
47 \( 1 + (126. + 126. i)T + 1.03e5iT^{2} \)
53 \( 1 + (254. - 254. i)T - 1.48e5iT^{2} \)
59 \( 1 + 131.T + 2.05e5T^{2} \)
61 \( 1 - 225.T + 2.26e5T^{2} \)
67 \( 1 + (-236. + 236. i)T - 3.00e5iT^{2} \)
71 \( 1 + 29.8iT - 3.57e5T^{2} \)
73 \( 1 + (41.4 + 41.4i)T + 3.89e5iT^{2} \)
79 \( 1 - 450. iT - 4.93e5T^{2} \)
83 \( 1 + (-729. + 729. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.47e3T + 7.04e5T^{2} \)
97 \( 1 + (-1.29e3 + 1.29e3i)T - 9.12e5iT^{2} \)
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

−11.11741254631489137623691780894, −10.42368296813288059392976828828, −9.499443738685823843244521273030, −8.472608867963263108611339361843, −7.40358192758289370860554412119, −5.95952845351495062372386947176, −5.11638476577352271834436889046, −3.44429427126015220973330490811, −1.79486601679705723193651714488, −0.35765007861849269709188670955, 2.06359812849108469866979716721, 3.64881860540044969377935878521, 4.96502216051185250075255527156, 6.37154366209821843393792273047, 7.53396851911172020121924221729, 8.108312488693065326807769965091, 9.355563130019008482907809743257, 10.02219201682971102272089893568, 11.64265484329112156811516962595, 12.16218765237614171141059648582

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