↑

Properties

Label 2-3240-9.7-c1-0-11
Degree 22
Conductor 32403240
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  + (0.5 + 0.866i)5-s + (1.18 − 2.05i)7-s + (−1.68 + 2.92i)11-s + (−1.18 − 2.05i)13-s − 4.74·17-s + 19-s + (−0.186 − 0.322i)23-s + (−0.499 + 0.866i)25-s + (−1.68 + 2.92i)29-s + (3.05 + 5.29i)31-s + 2.37·35-s + 6·37-s + (5.87 + 10.1i)41-s + (−3.37 + 5.84i)43-s + (−1.81 + 3.14i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.448 − 0.776i)7-s + (−0.508 + 0.880i)11-s + (−0.328 − 0.569i)13-s − 1.15·17-s + 0.229·19-s + (−0.0388 − 0.0672i)23-s + (−0.0999 + 0.173i)25-s + (−0.313 + 0.542i)29-s + (0.549 + 0.951i)31-s + 0.400·35-s + 0.986·37-s + (0.917 + 1.58i)41-s + (−0.514 + 0.890i)43-s + (−0.264 + 0.458i)47-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(2161,⋅)\chi_{3240} (2161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 0.173−0.984i)(2,\ 3240,\ (\ :1/2),\ 0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.4541976691.454197669
L(12)L(\frac12) ≈\approx 1.4541976691.454197669
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
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(−1.18+2.05i)T+(−3.5−6.06i)T2 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.68−2.92i)T+(−5.5−9.52i)T2 1 + (1.68 - 2.92i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.18+2.05i)T+(−6.5+11.2i)T2 1 + (1.18 + 2.05i)T + (-6.5 + 11.2i)T^{2}
17 1+4.74T+17T2 1 + 4.74T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+(0.186+0.322i)T+(−11.5+19.9i)T2 1 + (0.186 + 0.322i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.68−2.92i)T+(−14.5−25.1i)T2 1 + (1.68 - 2.92i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.05−5.29i)T+(−15.5+26.8i)T2 1 + (-3.05 - 5.29i)T + (-15.5 + 26.8i)T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+(−5.87−10.1i)T+(−20.5+35.5i)T2 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.37−5.84i)T+(−21.5−37.2i)T2 1 + (3.37 - 5.84i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.81−3.14i)T+(−23.5−40.7i)T2 1 + (1.81 - 3.14i)T + (-23.5 - 40.7i)T^{2}
53 1−7.11T+53T2 1 - 7.11T + 53T^{2}
59 1+(−2.5−4.33i)T+(−29.5+51.0i)T2 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.627−1.08i)T+(−30.5−52.8i)T2 1 + (0.627 - 1.08i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.37+9.30i)T+(−33.5+58.0i)T2 1 + (5.37 + 9.30i)T + (-33.5 + 58.0i)T^{2}
71 1+1.37T+71T2 1 + 1.37T + 71T^{2}
73 1−3.25T+73T2 1 - 3.25T + 73T^{2}
79 1+(4.37−7.57i)T+(−39.5−68.4i)T2 1 + (4.37 - 7.57i)T + (-39.5 - 68.4i)T^{2}
83 1+(5−8.66i)T+(−41.5−71.8i)T2 1 + (5 - 8.66i)T + (-41.5 - 71.8i)T^{2}
89 1+1.37T+89T2 1 + 1.37T + 89T^{2}
97 1+(3.37−5.84i)T+(−48.5−84.0i)T2 1 + (3.37 - 5.84i)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

−8.773071225898285748457803507611, −7.917670739725615947470090316414, −7.35974477052713977669017572903, −6.70422370775092627075043268479, −5.83353693111733644051869985713, −4.76410206688183916965380006555, −4.40710131673675750102138561264, −3.13493548238853908723297514829, −2.33345830853821436962046102734, −1.17321888833871824271831428920, 0.46400530550641295319183729903, 2.00340359272702501072677485001, 2.57703439589641857720868267979, 3.87908493669760562453600206740, 4.67692977067778736597453657844, 5.54105862305303005650400098581, 6.02252127372296267759680190717, 7.02019886011412688978816310250, 7.85372099658644272220522693457, 8.665465751586427530188620630444

Graph of the ZZ-function along the critical line