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Properties

Label 2-105-105.32-c1-0-6
Degree 22
Conductor 105105
Sign 0.770−0.637i0.770 - 0.637i
Analytic cond. 0.8384290.838429
Root an. cond. 0.9156570.915657
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  + (2.35 + 0.631i)2-s + (−1.54 + 0.775i)3-s + (3.42 + 1.97i)4-s + (−0.0540 − 2.23i)5-s + (−4.13 + 0.849i)6-s + (−1.91 + 1.82i)7-s + (3.36 + 3.36i)8-s + (1.79 − 2.40i)9-s + (1.28 − 5.30i)10-s + (−3.08 − 1.77i)11-s + (−6.83 − 0.406i)12-s + (1.28 − 1.28i)13-s + (−5.67 + 3.08i)14-s + (1.81 + 3.42i)15-s + (1.85 + 3.21i)16-s + (0.792 + 2.95i)17-s + ⋯
L(s)  = 1  + (1.66 + 0.446i)2-s + (−0.894 + 0.447i)3-s + (1.71 + 0.987i)4-s + (−0.0241 − 0.999i)5-s + (−1.68 + 0.346i)6-s + (−0.725 + 0.688i)7-s + (1.19 + 1.19i)8-s + (0.599 − 0.800i)9-s + (0.406 − 1.67i)10-s + (−0.928 − 0.536i)11-s + (−1.97 − 0.117i)12-s + (0.356 − 0.356i)13-s + (−1.51 + 0.823i)14-s + (0.469 + 0.883i)15-s + (0.463 + 0.803i)16-s + (0.192 + 0.717i)17-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=((0.770−0.637i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+1/2) L(s)=((0.770−0.637i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 105105    =    3⋅5⋅73 \cdot 5 \cdot 7
Sign: 0.770−0.637i0.770 - 0.637i
Analytic conductor: 0.8384290.838429
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ105(32,⋅)\chi_{105} (32, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 105, ( :1/2), 0.770−0.637i)(2,\ 105,\ (\ :1/2),\ 0.770 - 0.637i)

Particular Values

L(1)L(1) ≈\approx 1.62806+0.585717i1.62806 + 0.585717i
L(12)L(\frac12) ≈\approx 1.62806+0.585717i1.62806 + 0.585717i
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)
bad3 1+(1.54−0.775i)T 1 + (1.54 - 0.775i)T
5 1+(0.0540+2.23i)T 1 + (0.0540 + 2.23i)T
7 1+(1.91−1.82i)T 1 + (1.91 - 1.82i)T
good2 1+(−2.35−0.631i)T+(1.73+i)T2 1 + (-2.35 - 0.631i)T + (1.73 + i)T^{2}
11 1+(3.08+1.77i)T+(5.5+9.52i)T2 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2}
13 1+(−1.28+1.28i)T−13iT2 1 + (-1.28 + 1.28i)T - 13iT^{2}
17 1+(−0.792−2.95i)T+(−14.7+8.5i)T2 1 + (-0.792 - 2.95i)T + (-14.7 + 8.5i)T^{2}
19 1+(−0.331+0.191i)T+(9.5−16.4i)T2 1 + (-0.331 + 0.191i)T + (9.5 - 16.4i)T^{2}
23 1+(0.658−2.45i)T+(−19.9−11.5i)T2 1 + (0.658 - 2.45i)T + (-19.9 - 11.5i)T^{2}
29 1−5.51T+29T2 1 - 5.51T + 29T^{2}
31 1+(−0.323+0.561i)T+(−15.5−26.8i)T2 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.34−5.00i)T+(−32.0−18.5i)T2 1 + (1.34 - 5.00i)T + (-32.0 - 18.5i)T^{2}
41 1−10.1iT−41T2 1 - 10.1iT - 41T^{2}
43 1+(0.335−0.335i)T−43iT2 1 + (0.335 - 0.335i)T - 43iT^{2}
47 1+(−2.80−0.751i)T+(40.7+23.5i)T2 1 + (-2.80 - 0.751i)T + (40.7 + 23.5i)T^{2}
53 1+(3.04−0.815i)T+(45.8−26.5i)T2 1 + (3.04 - 0.815i)T + (45.8 - 26.5i)T^{2}
59 1+(−3.81+6.60i)T+(−29.5−51.0i)T2 1 + (-3.81 + 6.60i)T + (-29.5 - 51.0i)T^{2}
61 1+(5.45+9.45i)T+(−30.5+52.8i)T2 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2}
67 1+(−12.3+3.31i)T+(58.0−33.5i)T2 1 + (-12.3 + 3.31i)T + (58.0 - 33.5i)T^{2}
71 1+3.06iT−71T2 1 + 3.06iT - 71T^{2}
73 1+(0.849+3.17i)T+(−63.2+36.5i)T2 1 + (0.849 + 3.17i)T + (-63.2 + 36.5i)T^{2}
79 1+(3.21−1.85i)T+(39.5−68.4i)T2 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2}
83 1+(0.973+0.973i)T+83iT2 1 + (0.973 + 0.973i)T + 83iT^{2}
89 1+(−1.51−2.63i)T+(−44.5+77.0i)T2 1 + (-1.51 - 2.63i)T + (-44.5 + 77.0i)T^{2}
97 1+(10.3+10.3i)T+97iT2 1 + (10.3 + 10.3i)T + 97iT^{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

−13.60261886039178459235992854933, −12.80601086764414156635374259021, −12.25166257730708830343215997665, −11.20935306810921370871956843532, −9.780680426967857996457038678964, −8.180577503758516316971543172110, −6.38156973014447193831862892205, −5.63148947963667615192455781611, −4.77446497453703299763703119885, −3.35825673050670158052135747612, 2.57840070936464811318725115299, 4.11073559390851045737695272659, 5.47937492612628641895201661495, 6.60236395183664917839902528809, 7.31294601876405843772222570020, 10.21634446088090384657214377407, 10.76168491432123600880255555463, 11.82572555280642318453772648011, 12.66388240440585691984774421719, 13.58056449639412908991287292703

Graph of the ZZ-function along the critical line