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Properties

Label 2-72-72.59-c1-0-3
Degree 22
Conductor 7272
Sign 0.870−0.491i0.870 - 0.491i
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  + (−1.22 + 0.707i)2-s + (1.72 + 0.158i)3-s + (0.999 − 1.73i)4-s + (−2.22 + 1.02i)6-s + 2.82i·8-s + (2.94 + 0.548i)9-s + (−3.27 + 1.89i)11-s + (1.99 − 2.82i)12-s + (−2.00 − 3.46i)16-s − 8.02i·17-s + (−3.99 + 1.41i)18-s − 8.34·19-s + (2.67 − 4.63i)22-s + (−0.449 + 4.87i)24-s + (2.5 + 4.33i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.995 + 0.0917i)3-s + (0.499 − 0.866i)4-s + (−0.908 + 0.418i)6-s + 0.999i·8-s + (0.983 + 0.182i)9-s + (−0.987 + 0.570i)11-s + (0.577 − 0.816i)12-s + (−0.500 − 0.866i)16-s − 1.94i·17-s + (−0.942 + 0.333i)18-s − 1.91·19-s + (0.570 − 0.987i)22-s + (−0.0917 + 0.995i)24-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.870−0.491i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.870−0.491i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.870−0.491i0.870 - 0.491i
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), 0.870−0.491i)(2,\ 72,\ (\ :1/2),\ 0.870 - 0.491i)

Particular Values

L(1)L(1) ≈\approx 0.797740+0.209813i0.797740 + 0.209813i
L(12)L(\frac12) ≈\approx 0.797740+0.209813i0.797740 + 0.209813i
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+(1.22−0.707i)T 1 + (1.22 - 0.707i)T
3 1+(−1.72−0.158i)T 1 + (-1.72 - 0.158i)T
good5 1+(−2.5−4.33i)T2 1 + (-2.5 - 4.33i)T^{2}
7 1+(3.5−6.06i)T2 1 + (3.5 - 6.06i)T^{2}
11 1+(3.27−1.89i)T+(5.5−9.52i)T2 1 + (3.27 - 1.89i)T + (5.5 - 9.52i)T^{2}
13 1+(6.5+11.2i)T2 1 + (6.5 + 11.2i)T^{2}
17 1+8.02iT−17T2 1 + 8.02iT - 17T^{2}
19 1+8.34T+19T2 1 + 8.34T + 19T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(−14.5+25.1i)T2 1 + (-14.5 + 25.1i)T^{2}
31 1+(15.5+26.8i)T2 1 + (15.5 + 26.8i)T^{2}
37 1−37T2 1 - 37T^{2}
41 1+(0.398+0.230i)T+(20.5+35.5i)T2 1 + (0.398 + 0.230i)T + (20.5 + 35.5i)T^{2}
43 1+(−1.17−2.03i)T+(−21.5+37.2i)T2 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1+53T2 1 + 53T^{2}
59 1+(−10.6−6.13i)T+(29.5+51.0i)T2 1 + (-10.6 - 6.13i)T + (29.5 + 51.0i)T^{2}
61 1+(30.5−52.8i)T2 1 + (30.5 - 52.8i)T^{2}
67 1+(−7.17+12.4i)T+(−33.5−58.0i)T2 1 + (-7.17 + 12.4i)T + (-33.5 - 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1−13.6T+73T2 1 - 13.6T + 73T^{2}
79 1+(39.5−68.4i)T2 1 + (39.5 - 68.4i)T^{2}
83 1+(2.44−1.41i)T+(41.5−71.8i)T2 1 + (2.44 - 1.41i)T + (41.5 - 71.8i)T^{2}
89 1−5.65iT−89T2 1 - 5.65iT - 89T^{2}
97 1+(9.84+17.0i)T+(−48.5+84.0i)T2 1 + (9.84 + 17.0i)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

−14.99026046468207165228230224067, −13.94325831746413774485719409405, −12.75793173445158789758196750730, −11.01267020733973071648887699286, −9.918566536362943500092950394430, −8.976847986167529653157356874982, −7.86177047040072258534126108663, −6.88674705314451405041949624189, −4.91148107712993364417179104961, −2.46497567617430314622651811016, 2.26572919897296598094005540027, 3.89150805359828389086771997983, 6.56921485253876360296646337170, 8.193271439978103121820558346488, 8.542133080835188017557302649565, 10.12341874679207186463613675070, 10.82257133950445822006569975190, 12.59712718113872897415425981215, 13.13253616871827371677520951235, 14.68592575309039060882362215601

Graph of the ZZ-function along the critical line