↑

Properties

Label 2-45-5.3-c4-0-3
Degree $2$
Conductor $45$
Sign $0.525 - 0.850i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5 + 5i)2-s − 34i·4-s + 25·5-s + (40 − 40i)7-s + (90 + 90i)8-s + (−125 + 125i)10-s − 100·11-s + (205 + 205i)13-s + 400i·14-s − 356·16-s + (235 − 235i)17-s − 72i·19-s − 850i·20-s + (500 − 500i)22-s + (340 + 340i)23-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s − 2.12i·4-s + 5-s + (0.816 − 0.816i)7-s + (1.40 + 1.40i)8-s + (−1.25 + 1.25i)10-s − 0.826·11-s + (1.21 + 1.21i)13-s + 2.04i·14-s − 1.39·16-s + (0.813 − 0.813i)17-s − 0.199i·19-s − 2.12i·20-s + (1.03 − 1.03i)22-s + (0.642 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.884644 + 0.493221i\)
\(L(\frac12)\) \(\approx\) \(0.884644 + 0.493221i\)
\(L(3)\) 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 \( 1 - 25T \)
good2 \( 1 + (5 - 5i)T - 16iT^{2} \)
7 \( 1 + (-40 + 40i)T - 2.40e3iT^{2} \)
11 \( 1 + 100T + 1.46e4T^{2} \)
13 \( 1 + (-205 - 205i)T + 2.85e4iT^{2} \)
17 \( 1 + (-235 + 235i)T - 8.35e4iT^{2} \)
19 \( 1 + 72iT - 1.30e5T^{2} \)
23 \( 1 + (-340 - 340i)T + 2.79e5iT^{2} \)
29 \( 1 + 450iT - 7.07e5T^{2} \)
31 \( 1 - 428T + 9.23e5T^{2} \)
37 \( 1 + (755 - 755i)T - 1.87e6iT^{2} \)
41 \( 1 - 950T + 2.82e6T^{2} \)
43 \( 1 + (1.22e3 + 1.22e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (320 - 320i)T - 4.87e6iT^{2} \)
53 \( 1 + (-505 - 505i)T + 7.89e6iT^{2} \)
59 \( 1 + 6.30e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.80e3T + 1.38e7T^{2} \)
67 \( 1 + (-340 + 340i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.40e3T + 2.54e7T^{2} \)
73 \( 1 + (-415 - 415i)T + 2.83e7iT^{2} \)
79 \( 1 - 6.73e3iT - 3.89e7T^{2} \)
83 \( 1 + (680 + 680i)T + 4.74e7iT^{2} \)
89 \( 1 + 2.25e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.61e3 + 1.61e3i)T - 8.85e7iT^{2} \)
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

−15.60902321995802157718247358034, −14.22887761342581863350793123036, −13.60469477727313725774403377547, −11.13863414495670208643152394214, −10.05764281745735347802030116292, −8.978313391807621173776114886713, −7.74570646486791488147425204999, −6.56798480812039893955511375980, −5.18008670430117112402853184422, −1.29531835230208170270457325946, 1.39363567087694261239185012619, 2.88303186380730606771898343364, 5.61584908961168104185922907072, 8.063258574828759974350640315673, 8.842516389332419937877742541108, 10.27816425535131573361315562552, 10.87071480473455952488790464967, 12.32798336888523913000149918694, 13.23243016215544772796885912275, 14.93335667345749484841391268839

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