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Properties

Label 2-72-72.61-c1-0-6
Degree 22
Conductor 7272
Sign 0.819−0.573i0.819 - 0.573i
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  + (0.366 + 1.36i)2-s + (0.866 − 1.5i)3-s + (−1.73 + i)4-s + (1.73 − i)5-s + (2.36 + 0.633i)6-s + (−2 + 3.46i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (2 + 1.99i)10-s + (−2.59 − 1.5i)11-s + 3.46i·12-s + (−1.73 + i)13-s + (−5.46 − 1.46i)14-s − 3.46i·15-s + (1.99 − 3.46i)16-s + 5·17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.774 − 0.447i)5-s + (0.965 + 0.258i)6-s + (−0.755 + 1.30i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.632 + 0.632i)10-s + (−0.783 − 0.452i)11-s + 0.999i·12-s + (−0.480 + 0.277i)13-s + (−1.46 − 0.391i)14-s − 0.894i·15-s + (0.499 − 0.866i)16-s + 1.21·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.819−0.573i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.819−0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.819−0.573i0.819 - 0.573i
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), 0.819−0.573i)(2,\ 72,\ (\ :1/2),\ 0.819 - 0.573i)

Particular Values

L(1)L(1) ≈\approx 1.04333+0.328961i1.04333 + 0.328961i
L(12)L(\frac12) ≈\approx 1.04333+0.328961i1.04333 + 0.328961i
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+(−0.366−1.36i)T 1 + (-0.366 - 1.36i)T
3 1+(−0.866+1.5i)T 1 + (-0.866 + 1.5i)T
good5 1+(−1.73+i)T+(2.5−4.33i)T2 1 + (-1.73 + i)T + (2.5 - 4.33i)T^{2}
7 1+(2−3.46i)T+(−3.5−6.06i)T2 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.59+1.5i)T+(5.5+9.52i)T2 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2}
13 1+(1.73−i)T+(6.5−11.2i)T2 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2}
17 1−5T+17T2 1 - 5T + 17T^{2}
19 1+iT−19T2 1 + iT - 19T^{2}
23 1+(1+1.73i)T+(−11.5+19.9i)T2 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2}
29 1+(14.5+25.1i)T2 1 + (14.5 + 25.1i)T^{2}
31 1+(−2−3.46i)T+(−15.5+26.8i)T2 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+(−2.5−4.33i)T+(−20.5+35.5i)T2 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2}
43 1+(−9.52−5.5i)T+(21.5+37.2i)T2 1 + (-9.52 - 5.5i)T + (21.5 + 37.2i)T^{2}
47 1+(−3+5.19i)T+(−23.5−40.7i)T2 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}
53 1−53T2 1 - 53T^{2}
59 1+(−0.866+0.5i)T+(29.5−51.0i)T2 1 + (-0.866 + 0.5i)T + (29.5 - 51.0i)T^{2}
61 1+(10.3+6i)T+(30.5+52.8i)T2 1 + (10.3 + 6i)T + (30.5 + 52.8i)T^{2}
67 1+(2.59−1.5i)T+(33.5−58.0i)T2 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−9T+73T2 1 - 9T + 73T^{2}
79 1+(−7+12.1i)T+(−39.5−68.4i)T2 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.46+2i)T+(41.5+71.8i)T2 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1+(0.5−0.866i)T+(−48.5−84.0i)T2 1 + (0.5 - 0.866i)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

−14.65225774412273451644233915024, −13.70206001225455086377290891423, −12.77753997839591242264394960293, −12.18207237333949929815455512245, −9.640827493928599083202869088148, −8.864233791200871868373160028945, −7.72766623118291055585766666734, −6.24321063915552237522606768556, −5.44469430329803475229840398074, −2.87052640885778066793988540552, 2.75148535640928773077850167267, 4.08380052048807056291574053817, 5.62945459857429237262547075789, 7.69991990660725264961026418096, 9.557197029839213090619546303397, 10.13251765185218936814101371061, 10.72570299456177282736360006327, 12.49059798390126917928632934601, 13.67445576818653449017017779628, 14.13727068886058197892349261299

Graph of the ZZ-function along the critical line