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Properties

Label 2-21-7.2-c3-0-1
Degree 22
Conductor 2121
Sign 0.880−0.473i0.880 - 0.473i
Analytic cond. 1.239041.23904
Root an. cond. 1.113121.11312
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.124 + 0.214i)2-s + (1.5 + 2.59i)3-s + (3.96 + 6.87i)4-s + (6.21 − 10.7i)5-s − 0.744·6-s + (−18.4 − 1.73i)7-s − 3.95·8-s + (−4.5 + 7.79i)9-s + (1.54 + 2.67i)10-s + (−30.1 − 52.2i)11-s + (−11.9 + 20.6i)12-s + 36.4·13-s + (2.66 − 3.74i)14-s + 37.3·15-s + (−31.2 + 54.1i)16-s + (24.3 + 42.2i)17-s + ⋯
L(s)  = 1  + (−0.0438 + 0.0759i)2-s + (0.288 + 0.499i)3-s + (0.496 + 0.859i)4-s + (0.556 − 0.963i)5-s − 0.0506·6-s + (−0.995 − 0.0938i)7-s − 0.174·8-s + (−0.166 + 0.288i)9-s + (0.0487 + 0.0844i)10-s + (−0.826 − 1.43i)11-s + (−0.286 + 0.496i)12-s + 0.777·13-s + (0.0507 − 0.0715i)14-s + 0.642·15-s + (−0.488 + 0.846i)16-s + (0.347 + 0.602i)17-s + ⋯

Functional equation

Λ(s)=(21s/2 ΓC(s) L(s)=((0.880−0.473i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(21s/2 ΓC(s+3/2) L(s)=((0.880−0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2121    =    3⋅73 \cdot 7
Sign: 0.880−0.473i0.880 - 0.473i
Analytic conductor: 1.239041.23904
Root analytic conductor: 1.113121.11312
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ21(16,⋅)\chi_{21} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 21, ( :3/2), 0.880−0.473i)(2,\ 21,\ (\ :3/2),\ 0.880 - 0.473i)

Particular Values

L(2)L(2) ≈\approx 1.18131+0.297209i1.18131 + 0.297209i
L(12)L(\frac12) ≈\approx 1.18131+0.297209i1.18131 + 0.297209i
L(52)L(\frac{5}{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)
bad3 1+(−1.5−2.59i)T 1 + (-1.5 - 2.59i)T
7 1+(18.4+1.73i)T 1 + (18.4 + 1.73i)T
good2 1+(0.124−0.214i)T+(−4−6.92i)T2 1 + (0.124 - 0.214i)T + (-4 - 6.92i)T^{2}
5 1+(−6.21+10.7i)T+(−62.5−108.i)T2 1 + (-6.21 + 10.7i)T + (-62.5 - 108. i)T^{2}
11 1+(30.1+52.2i)T+(−665.5+1.15e3i)T2 1 + (30.1 + 52.2i)T + (-665.5 + 1.15e3i)T^{2}
13 1−36.4T+2.19e3T2 1 - 36.4T + 2.19e3T^{2}
17 1+(−24.3−42.2i)T+(−2.45e3+4.25e3i)T2 1 + (-24.3 - 42.2i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−25.2+43.7i)T+(−3.42e3−5.94e3i)T2 1 + (-25.2 + 43.7i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(69.3−120.i)T+(−6.08e3−1.05e4i)T2 1 + (69.3 - 120. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+61.1T+2.43e4T2 1 + 61.1T + 2.43e4T^{2}
31 1+(−0.584−1.01i)T+(−1.48e4+2.57e4i)T2 1 + (-0.584 - 1.01i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(34.7−60.2i)T+(−2.53e4−4.38e4i)T2 1 + (34.7 - 60.2i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−308.T+6.89e4T2 1 - 308.T + 6.89e4T^{2}
43 1−174.T+7.95e4T2 1 - 174.T + 7.95e4T^{2}
47 1+(194.−337.i)T+(−5.19e4−8.99e4i)T2 1 + (194. - 337. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(157.+272.i)T+(−7.44e4+1.28e5i)T2 1 + (157. + 272. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(422.+731.i)T+(−1.02e5+1.77e5i)T2 1 + (422. + 731. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−169.+293.i)T+(−1.13e5−1.96e5i)T2 1 + (-169. + 293. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−485.−841.i)T+(−1.50e5+2.60e5i)T2 1 + (-485. - 841. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+98.4T+3.57e5T2 1 + 98.4T + 3.57e5T^{2}
73 1+(355.+615.i)T+(−1.94e5+3.36e5i)T2 1 + (355. + 615. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−243.+421.i)T+(−2.46e5−4.26e5i)T2 1 + (-243. + 421. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−605.T+5.71e5T2 1 - 605.T + 5.71e5T^{2}
89 1+(109.−188.i)T+(−3.52e5−6.10e5i)T2 1 + (109. - 188. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+782.T+9.12e5T2 1 + 782.T + 9.12e5T^{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

−17.42840234217564586993226542844, −16.23055622637122962884479404939, −15.89661175917865023681722252124, −13.58223739559200347317134805619, −12.82073607332888322503371272846, −11.05263346182170649797912524248, −9.332721178444415412381209580040, −8.119892828999079687461125020766, −5.88817185904019278055773599757, −3.38228381260687071477557465693, 2.48372776722578918437887004989, 6.03416664316828508044427140253, 7.18989616090266510259301698319, 9.670961843150247179522507042355, 10.57294761895014104508982770498, 12.40634587593307672436147169363, 13.85550663964881382335246218730, 14.94093638176937574859696846384, 16.09129734937073203102080876666, 18.18892700008772531938131117749

Graph of the ZZ-function along the critical line