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Properties

Label 2-72-72.61-c1-0-1
Degree 22
Conductor 7272
Sign −0.130−0.991i-0.130 - 0.991i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.179 + 1.40i)2-s + (0.986 + 1.42i)3-s + (−1.93 − 0.504i)4-s + (−1.19 + 0.687i)5-s + (−2.17 + 1.12i)6-s + (1.80 − 3.12i)7-s + (1.05 − 2.62i)8-s + (−1.05 + 2.80i)9-s + (−0.750 − 1.79i)10-s + (1.83 + 1.05i)11-s + (−1.19 − 3.25i)12-s + (−0.887 + 0.512i)13-s + (4.06 + 3.09i)14-s + (−2.15 − 1.01i)15-s + (3.49 + 1.95i)16-s + 0.808·17-s + ⋯
L(s)  = 1  + (−0.127 + 0.991i)2-s + (0.569 + 0.822i)3-s + (−0.967 − 0.252i)4-s + (−0.532 + 0.307i)5-s + (−0.887 + 0.460i)6-s + (0.682 − 1.18i)7-s + (0.373 − 0.927i)8-s + (−0.351 + 0.936i)9-s + (−0.237 − 0.567i)10-s + (0.552 + 0.319i)11-s + (−0.343 − 0.939i)12-s + (−0.246 + 0.142i)13-s + (1.08 + 0.826i)14-s + (−0.556 − 0.262i)15-s + (0.872 + 0.487i)16-s + 0.196·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.130−0.991i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((−0.130−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.130−0.991i-0.130 - 0.991i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(61,⋅)\chi_{72} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), −0.130−0.991i)(2,\ 72,\ (\ :1/2),\ -0.130 - 0.991i)

Particular Values

L(1)L(1) ≈\approx 0.603924+0.688632i0.603924 + 0.688632i
L(12)L(\frac12) ≈\approx 0.603924+0.688632i0.603924 + 0.688632i
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+(0.179−1.40i)T 1 + (0.179 - 1.40i)T
3 1+(−0.986−1.42i)T 1 + (-0.986 - 1.42i)T
good5 1+(1.19−0.687i)T+(2.5−4.33i)T2 1 + (1.19 - 0.687i)T + (2.5 - 4.33i)T^{2}
7 1+(−1.80+3.12i)T+(−3.5−6.06i)T2 1 + (-1.80 + 3.12i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.83−1.05i)T+(5.5+9.52i)T2 1 + (-1.83 - 1.05i)T + (5.5 + 9.52i)T^{2}
13 1+(0.887−0.512i)T+(6.5−11.2i)T2 1 + (0.887 - 0.512i)T + (6.5 - 11.2i)T^{2}
17 1−0.808T+17T2 1 - 0.808T + 17T^{2}
19 1+7.43iT−19T2 1 + 7.43iT - 19T^{2}
23 1+(1.65+2.86i)T+(−11.5+19.9i)T2 1 + (1.65 + 2.86i)T + (-11.5 + 19.9i)T^{2}
29 1+(7.71+4.45i)T+(14.5+25.1i)T2 1 + (7.71 + 4.45i)T + (14.5 + 25.1i)T^{2}
31 1+(−3.26−5.65i)T+(−15.5+26.8i)T2 1 + (-3.26 - 5.65i)T + (-15.5 + 26.8i)T^{2}
37 1−4.01iT−37T2 1 - 4.01iT - 37T^{2}
41 1+(3.45+5.99i)T+(−20.5+35.5i)T2 1 + (3.45 + 5.99i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.245+0.142i)T+(21.5+37.2i)T2 1 + (0.245 + 0.142i)T + (21.5 + 37.2i)T^{2}
47 1+(3.61−6.25i)T+(−23.5−40.7i)T2 1 + (3.61 - 6.25i)T + (-23.5 - 40.7i)T^{2}
53 1−3.86iT−53T2 1 - 3.86iT - 53T^{2}
59 1+(−7.06+4.08i)T+(29.5−51.0i)T2 1 + (-7.06 + 4.08i)T + (29.5 - 51.0i)T^{2}
61 1+(−6.31−3.64i)T+(30.5+52.8i)T2 1 + (-6.31 - 3.64i)T + (30.5 + 52.8i)T^{2}
67 1+(2.43−1.40i)T+(33.5−58.0i)T2 1 + (2.43 - 1.40i)T + (33.5 - 58.0i)T^{2}
71 1−4.69T+71T2 1 - 4.69T + 71T^{2}
73 1−0.409T+73T2 1 - 0.409T + 73T^{2}
79 1+(0.0456−0.0790i)T+(−39.5−68.4i)T2 1 + (0.0456 - 0.0790i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.40+1.39i)T+(41.5+71.8i)T2 1 + (2.40 + 1.39i)T + (41.5 + 71.8i)T^{2}
89 1−8.91T+89T2 1 - 8.91T + 89T^{2}
97 1+(2.76−4.78i)T+(−48.5−84.0i)T2 1 + (2.76 - 4.78i)T + (-48.5 - 84.0i)T^{2}
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   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

−14.98171010821158257131000638943, −14.21737514296421696754140492238, −13.35274119146532110058307530158, −11.35090225813189974519359573098, −10.24334137642268194022638861089, −9.107512481571810614785397136960, −7.88492660428633097179564135335, −6.98719425789938721107315552472, −4.87546041751562841196239980775, −3.90102024084730398680336707285, 1.90477263585986738383773588277, 3.68930295527200324136076743947, 5.65338999169909647334177879957, 7.88387243578902462271828246565, 8.539703991784501036174565693407, 9.687675386685760306925843625451, 11.55877124551154408946280671403, 12.01962611044808486056117478712, 12.96361828520117407246811989733, 14.25882852741799366713920528241

Graph of the ZZ-function along the critical line