↑

Properties

Label 2-4320-72.11-c1-0-19
Degree $2$
Conductor $4320$
Sign $0.909 - 0.415i$
Analytic cond. $34.4953$
Root an. cond. $5.87327$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (2.20 + 1.27i)7-s + (−4.02 − 2.32i)11-s + (3.23 − 1.87i)13-s + 3.38i·17-s + 5.12·19-s + (4.16 + 7.21i)23-s + (−0.499 + 0.866i)25-s + (−2.01 + 3.49i)29-s + (−4.88 + 2.81i)31-s − 2.54i·35-s − 7.40i·37-s + (5.23 − 3.02i)41-s + (−5.26 + 9.11i)43-s + (0.621 − 1.07i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.834 + 0.481i)7-s + (−1.21 − 0.701i)11-s + (0.898 − 0.518i)13-s + 0.821i·17-s + 1.17·19-s + (0.868 + 1.50i)23-s + (−0.0999 + 0.173i)25-s + (−0.375 + 0.649i)29-s + (−0.876 + 0.506i)31-s − 0.430i·35-s − 1.21i·37-s + (0.817 − 0.471i)41-s + (−0.802 + 1.39i)43-s + (0.0905 − 0.156i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4320\)    =    \(2^{5} \cdot 3^{3} \cdot 5\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(34.4953\)
Root analytic conductor: \(5.87327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4320} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4320,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.939452017\)
\(L(\frac12)\) \(\approx\) \(1.939452017\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-2.20 - 1.27i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.02 + 2.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.23 + 1.87i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.38iT - 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
23 \( 1 + (-4.16 - 7.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.01 - 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.88 - 2.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 + (-5.23 + 3.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.26 - 9.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.621 + 1.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + (0.351 - 0.202i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.57 + 2.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.17 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.43T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + (5.40 + 3.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.0 - 6.97i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.72iT - 89T^{2} \)
97 \( 1 + (-1.31 + 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.344165416394656732119990906089, −7.83168644554071972526611541303, −7.21466259306667413749403641266, −5.92315289974981801051818745808, −5.41634605214945317062720255140, −4.98780232597244479916604626107, −3.67163201766695680015740480261, −3.16144596090224134632693147476, −1.89213058993676151841850732335, −0.958211141563513986988745583676, 0.68114186761425515130705691867, 1.93059239043780028790525368354, 2.82727360274870612175672176287, 3.76035346441636897930168166429, 4.75578205651516663800961880216, 5.09948772541151846222121776085, 6.22603092315472067204738652887, 7.01899212504159602239919795131, 7.64532376998975788134175988217, 8.108377743824751447094638283921

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