↑

Properties

Label 2-3024-252.103-c1-0-32
Degree 22
Conductor 30243024
Sign 0.0138+0.999i0.0138 + 0.999i
Analytic cond. 24.146724.1467
Root an. cond. 4.913934.91393
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.243 − 0.140i)5-s + (−2.20 − 1.46i)7-s + (−2.44 − 1.41i)11-s + (4.06 + 2.34i)13-s + (7.00 − 4.04i)17-s + (0.474 − 0.821i)19-s + (−0.339 + 0.196i)23-s + (−2.46 + 4.26i)25-s + (−1.51 − 2.61i)29-s + 2.12·31-s + (−0.741 − 0.0462i)35-s + (−2.43 + 4.21i)37-s + (−0.478 − 0.276i)41-s + (−4.28 + 2.47i)43-s − 2.78·47-s + ⋯
L(s)  = 1  + (0.108 − 0.0627i)5-s + (−0.833 − 0.552i)7-s + (−0.736 − 0.425i)11-s + (1.12 + 0.651i)13-s + (1.69 − 0.980i)17-s + (0.108 − 0.188i)19-s + (−0.0708 + 0.0409i)23-s + (−0.492 + 0.852i)25-s + (−0.280 − 0.486i)29-s + 0.382·31-s + (−0.125 − 0.00781i)35-s + (−0.400 + 0.693i)37-s + (−0.0748 − 0.0431i)41-s + (−0.653 + 0.377i)43-s − 0.406·47-s + ⋯

Functional equation

Λ(s)=(3024s/2 ΓC(s) L(s)=((0.0138+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3024s/2 ΓC(s+1/2) L(s)=((0.0138+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30243024    =    24⋅33⋅72^{4} \cdot 3^{3} \cdot 7
Sign: 0.0138+0.999i0.0138 + 0.999i
Analytic conductor: 24.146724.1467
Root analytic conductor: 4.913934.91393
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3024(1279,⋅)\chi_{3024} (1279, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3024, ( :1/2), 0.0138+0.999i)(2,\ 3024,\ (\ :1/2),\ 0.0138 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.3843281931.384328193
L(12)L(\frac12) ≈\approx 1.3843281931.384328193
L(32)L(\frac{3}{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 1
7 1+(2.20+1.46i)T 1 + (2.20 + 1.46i)T
good5 1+(−0.243+0.140i)T+(2.5−4.33i)T2 1 + (-0.243 + 0.140i)T + (2.5 - 4.33i)T^{2}
11 1+(2.44+1.41i)T+(5.5+9.52i)T2 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2}
13 1+(−4.06−2.34i)T+(6.5+11.2i)T2 1 + (-4.06 - 2.34i)T + (6.5 + 11.2i)T^{2}
17 1+(−7.00+4.04i)T+(8.5−14.7i)T2 1 + (-7.00 + 4.04i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.474+0.821i)T+(−9.5−16.4i)T2 1 + (-0.474 + 0.821i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.339−0.196i)T+(11.5−19.9i)T2 1 + (0.339 - 0.196i)T + (11.5 - 19.9i)T^{2}
29 1+(1.51+2.61i)T+(−14.5+25.1i)T2 1 + (1.51 + 2.61i)T + (-14.5 + 25.1i)T^{2}
31 1−2.12T+31T2 1 - 2.12T + 31T^{2}
37 1+(2.43−4.21i)T+(−18.5−32.0i)T2 1 + (2.43 - 4.21i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.478+0.276i)T+(20.5+35.5i)T2 1 + (0.478 + 0.276i)T + (20.5 + 35.5i)T^{2}
43 1+(4.28−2.47i)T+(21.5−37.2i)T2 1 + (4.28 - 2.47i)T + (21.5 - 37.2i)T^{2}
47 1+2.78T+47T2 1 + 2.78T + 47T^{2}
53 1+(6.21+10.7i)T+(−26.5+45.8i)T2 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2}
59 1−11.4T+59T2 1 - 11.4T + 59T^{2}
61 1+11.1iT−61T2 1 + 11.1iT - 61T^{2}
67 1+9.07iT−67T2 1 + 9.07iT - 67T^{2}
71 1+1.54iT−71T2 1 + 1.54iT - 71T^{2}
73 1+(0.542−0.313i)T+(36.5−63.2i)T2 1 + (0.542 - 0.313i)T + (36.5 - 63.2i)T^{2}
79 1+15.4iT−79T2 1 + 15.4iT - 79T^{2}
83 1+(5.30+9.19i)T+(−41.5+71.8i)T2 1 + (5.30 + 9.19i)T + (-41.5 + 71.8i)T^{2}
89 1+(9.90+5.71i)T+(44.5+77.0i)T2 1 + (9.90 + 5.71i)T + (44.5 + 77.0i)T^{2}
97 1+(−13.6+7.86i)T+(48.5−84.0i)T2 1 + (-13.6 + 7.86i)T + (48.5 - 84.0i)T^{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

−8.461328782071546993908974724571, −7.79691447584164230282053603773, −7.03450327742547144775732447771, −6.25210327801493099917928842868, −5.54676416848434623389337042776, −4.69741291428179515898598487330, −3.44279450923308829278768877325, −3.21411929682989643960839388615, −1.68487591146575881543419791219, −0.47975206555921238289392700455, 1.17009717062827637818257306261, 2.44134413032670403458142816785, 3.32663626236048327776024595987, 4.00394279279654985557507936603, 5.41651851633680133973616334426, 5.73805395590846554418029826170, 6.52547964647405533618466273540, 7.49923128310235814499308563565, 8.229527396233165004214066388097, 8.759821812544840003416044023726

Graph of the ZZ-function along the critical line