↑

Properties

Label 2-72-72.59-c1-0-0
Degree 22
Conductor 7272
Sign −0.932−0.360i-0.932 - 0.360i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
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  + (−0.608 + 1.27i)2-s + (−1.71 + 0.231i)3-s + (−1.25 − 1.55i)4-s + (−1.74 + 3.01i)5-s + (0.748 − 2.33i)6-s + (−1.80 + 1.04i)7-s + (2.75 − 0.660i)8-s + (2.89 − 0.795i)9-s + (−2.79 − 4.06i)10-s + (−0.116 + 0.0675i)11-s + (2.52 + 2.37i)12-s + (2.63 + 1.52i)13-s + (−0.231 − 2.94i)14-s + (2.29 − 5.58i)15-s + (−0.830 + 3.91i)16-s + 4.19i·17-s + ⋯
L(s)  = 1  + (−0.430 + 0.902i)2-s + (−0.990 + 0.133i)3-s + (−0.629 − 0.777i)4-s + (−0.779 + 1.35i)5-s + (0.305 − 0.952i)6-s + (−0.683 + 0.394i)7-s + (0.972 − 0.233i)8-s + (0.964 − 0.265i)9-s + (−0.883 − 1.28i)10-s + (−0.0352 + 0.0203i)11-s + (0.727 + 0.685i)12-s + (0.731 + 0.422i)13-s + (−0.0619 − 0.786i)14-s + (0.591 − 1.44i)15-s + (−0.207 + 0.978i)16-s + 1.01i·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.932−0.360i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((−0.932−0.360i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.932−0.360i-0.932 - 0.360i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(59,⋅)\chi_{72} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), −0.932−0.360i)(2,\ 72,\ (\ :1/2),\ -0.932 - 0.360i)

Particular Values

L(1)L(1) ≈\approx 0.0734457+0.394335i0.0734457 + 0.394335i
L(12)L(\frac12) ≈\approx 0.0734457+0.394335i0.0734457 + 0.394335i
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+(0.608−1.27i)T 1 + (0.608 - 1.27i)T
3 1+(1.71−0.231i)T 1 + (1.71 - 0.231i)T
good5 1+(1.74−3.01i)T+(−2.5−4.33i)T2 1 + (1.74 - 3.01i)T + (-2.5 - 4.33i)T^{2}
7 1+(1.80−1.04i)T+(3.5−6.06i)T2 1 + (1.80 - 1.04i)T + (3.5 - 6.06i)T^{2}
11 1+(0.116−0.0675i)T+(5.5−9.52i)T2 1 + (0.116 - 0.0675i)T + (5.5 - 9.52i)T^{2}
13 1+(−2.63−1.52i)T+(6.5+11.2i)T2 1 + (-2.63 - 1.52i)T + (6.5 + 11.2i)T^{2}
17 1−4.19iT−17T2 1 - 4.19iT - 17T^{2}
19 1−0.919T+19T2 1 - 0.919T + 19T^{2}
23 1+(0.689−1.19i)T+(−11.5−19.9i)T2 1 + (0.689 - 1.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.24+7.34i)T+(−14.5+25.1i)T2 1 + (4.24 + 7.34i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.39−2.53i)T+(15.5+26.8i)T2 1 + (-4.39 - 2.53i)T + (15.5 + 26.8i)T^{2}
37 1+1.61iT−37T2 1 + 1.61iT - 37T^{2}
41 1+(−1.79−1.03i)T+(20.5+35.5i)T2 1 + (-1.79 - 1.03i)T + (20.5 + 35.5i)T^{2}
43 1+(−5.41−9.37i)T+(−21.5+37.2i)T2 1 + (-5.41 - 9.37i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.205−0.356i)T+(−23.5+40.7i)T2 1 + (-0.205 - 0.356i)T + (-23.5 + 40.7i)T^{2}
53 1−0.968T+53T2 1 - 0.968T + 53T^{2}
59 1+(−3.88−2.24i)T+(29.5+51.0i)T2 1 + (-3.88 - 2.24i)T + (29.5 + 51.0i)T^{2}
61 1+(−7.44+4.29i)T+(30.5−52.8i)T2 1 + (-7.44 + 4.29i)T + (30.5 - 52.8i)T^{2}
67 1+(−3.15+5.46i)T+(−33.5−58.0i)T2 1 + (-3.15 + 5.46i)T + (-33.5 - 58.0i)T^{2}
71 1−11.9T+71T2 1 - 11.9T + 71T^{2}
73 1+4.06T+73T2 1 + 4.06T + 73T^{2}
79 1+(10.8−6.27i)T+(39.5−68.4i)T2 1 + (10.8 - 6.27i)T + (39.5 - 68.4i)T^{2}
83 1+(−5.23+3.02i)T+(41.5−71.8i)T2 1 + (-5.23 + 3.02i)T + (41.5 - 71.8i)T^{2}
89 1+8.35iT−89T2 1 + 8.35iT - 89T^{2}
97 1+(0.477+0.826i)T+(−48.5+84.0i)T2 1 + (0.477 + 0.826i)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

−15.51696265619165483476802732301, −14.45189246662506312492286272234, −13.01272984312161346567660095415, −11.52697555067362037596343642602, −10.65620622063244349373937613232, −9.596736430884011093391556177854, −7.88534332612652737367525116376, −6.66940874963970156513598629788, −5.97572994778091420032943754136, −3.98966305529048461110879331557, 0.73549784888568791311413908671, 3.90969286045348549593597225416, 5.20723990760421266340880355064, 7.27115535437815740836172397684, 8.587903196434985901034352347965, 9.792387384558219904766919260241, 11.01261926034738595552759382896, 11.96672221569593507097049883167, 12.74184592215779855643027629883, 13.49394757874877752798320310684

Graph of the ZZ-function along the critical line