↑

Properties

Label 2-240-15.14-c2-0-1
Degree 22
Conductor 240240
Sign −0.541−0.840i-0.541 - 0.840i
Analytic cond. 6.539526.53952
Root an. cond. 2.557242.55724
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 + 1.26i)3-s + (−0.689 − 4.95i)5-s + 0.735i·7-s + (5.82 − 6.86i)9-s + 10.9i·11-s + 21.1i·13-s + (8.11 + 12.6i)15-s − 7.03·17-s − 23.1·19-s + (−0.927 − 2.00i)21-s − 24.7·23-s + (−24.0 + 6.82i)25-s + (−7.21 + 26.0i)27-s + 32.3i·29-s + 34.9·31-s + ⋯
L(s)  = 1  + (−0.907 + 0.420i)3-s + (−0.137 − 0.990i)5-s + 0.105i·7-s + (0.647 − 0.762i)9-s + 0.995i·11-s + 1.63i·13-s + (0.541 + 0.840i)15-s − 0.413·17-s − 1.21·19-s + (−0.0441 − 0.0953i)21-s − 1.07·23-s + (−0.962 + 0.272i)25-s + (−0.267 + 0.963i)27-s + 1.11i·29-s + 1.12·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.541−0.840i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+1) L(s)=((−0.541−0.840i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.541−0.840i-0.541 - 0.840i
Analytic conductor: 6.539526.53952
Root analytic conductor: 2.557242.55724
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ240(209,⋅)\chi_{240} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :1), −0.541−0.840i)(2,\ 240,\ (\ :1),\ -0.541 - 0.840i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.279947+0.512987i0.279947 + 0.512987i
L(12)L(\frac12) ≈\approx 0.279947+0.512987i0.279947 + 0.512987i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(2.72−1.26i)T 1 + (2.72 - 1.26i)T
5 1+(0.689+4.95i)T 1 + (0.689 + 4.95i)T
good7 1−0.735iT−49T2 1 - 0.735iT - 49T^{2}
11 1−10.9iT−121T2 1 - 10.9iT - 121T^{2}
13 1−21.1iT−169T2 1 - 21.1iT - 169T^{2}
17 1+7.03T+289T2 1 + 7.03T + 289T^{2}
19 1+23.1T+361T2 1 + 23.1T + 361T^{2}
23 1+24.7T+529T2 1 + 24.7T + 529T^{2}
29 1−32.3iT−841T2 1 - 32.3iT - 841T^{2}
31 1−34.9T+961T2 1 - 34.9T + 961T^{2}
37 1−37.7iT−1.36e3T2 1 - 37.7iT - 1.36e3T^{2}
41 1−39.0iT−1.68e3T2 1 - 39.0iT - 1.68e3T^{2}
43 1+22.6iT−1.84e3T2 1 + 22.6iT - 1.84e3T^{2}
47 1−39.1T+2.20e3T2 1 - 39.1T + 2.20e3T^{2}
53 1+60.9T+2.80e3T2 1 + 60.9T + 2.80e3T^{2}
59 1−7.79iT−3.48e3T2 1 - 7.79iT - 3.48e3T^{2}
61 1+11.1T+3.72e3T2 1 + 11.1T + 3.72e3T^{2}
67 1+33.3iT−4.48e3T2 1 + 33.3iT - 4.48e3T^{2}
71 1−96.9iT−5.04e3T2 1 - 96.9iT - 5.04e3T^{2}
73 1+134.iT−5.32e3T2 1 + 134. iT - 5.32e3T^{2}
79 1+121.T+6.24e3T2 1 + 121.T + 6.24e3T^{2}
83 1+90.2T+6.88e3T2 1 + 90.2T + 6.88e3T^{2}
89 1−53.1iT−7.92e3T2 1 - 53.1iT - 7.92e3T^{2}
97 1+115.iT−9.40e3T2 1 + 115. iT - 9.40e3T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.13554300568798031519158291897, −11.47724190951364477722410529501, −10.27015273488240181560640990603, −9.410461883244732884124003140507, −8.530183334357071432734297520044, −7.02576196381714374124277524675, −6.09163876736711514637891654026, −4.66705342601054466290572088158, −4.26608441468320276497056817350, −1.70886955074925118310841616316, 0.34285760598174426797576369275, 2.52660127541498326016691504960, 4.07496334669074033529700467753, 5.73500903592233743477642005048, 6.30221189908955298440743776005, 7.52405642002791425299283162094, 8.333838305590120229492029915052, 10.13906903101663305075342110310, 10.68387717761609215626701490395, 11.44127694068565107479016454341

Graph of the ZZ-function along the critical line