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Properties

Label 2-2160-9.7-c1-0-13
Degree 22
Conductor 21602160
Sign 0.984+0.173i0.984 + 0.173i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
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  + (−0.5 − 0.866i)5-s + (0.133 − 0.232i)7-s + (0.732 − 1.26i)11-s + (2.73 + 4.73i)13-s − 0.535·17-s + 2·19-s + (−1.86 − 3.23i)23-s + (−0.499 + 0.866i)25-s + (0.767 − 1.33i)29-s + (−1 − 1.73i)31-s − 0.267·35-s + 10.3·37-s + (4.96 + 8.59i)41-s + (−2.26 + 3.92i)43-s + (−0.133 + 0.232i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.0506 − 0.0877i)7-s + (0.220 − 0.382i)11-s + (0.757 + 1.31i)13-s − 0.129·17-s + 0.458·19-s + (−0.389 − 0.673i)23-s + (−0.0999 + 0.173i)25-s + (0.142 − 0.246i)29-s + (−0.179 − 0.311i)31-s − 0.0452·35-s + 1.70·37-s + (0.775 + 1.34i)41-s + (−0.345 + 0.599i)43-s + (−0.0195 + 0.0338i)47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.984+0.173i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1/2) L(s)=((0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.984+0.173i0.984 + 0.173i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(721,⋅)\chi_{2160} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), 0.984+0.173i)(2,\ 2160,\ (\ :1/2),\ 0.984 + 0.173i)

Particular Values

L(1)L(1) ≈\approx 1.7737909001.773790900
L(12)L(\frac12) ≈\approx 1.7737909001.773790900
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
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−0.133+0.232i)T+(−3.5−6.06i)T2 1 + (-0.133 + 0.232i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.732+1.26i)T+(−5.5−9.52i)T2 1 + (-0.732 + 1.26i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.73−4.73i)T+(−6.5+11.2i)T2 1 + (-2.73 - 4.73i)T + (-6.5 + 11.2i)T^{2}
17 1+0.535T+17T2 1 + 0.535T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+(1.86+3.23i)T+(−11.5+19.9i)T2 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.767+1.33i)T+(−14.5−25.1i)T2 1 + (-0.767 + 1.33i)T + (-14.5 - 25.1i)T^{2}
31 1+(1+1.73i)T+(−15.5+26.8i)T2 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1+(−4.96−8.59i)T+(−20.5+35.5i)T2 1 + (-4.96 - 8.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.26−3.92i)T+(−21.5−37.2i)T2 1 + (2.26 - 3.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.133−0.232i)T+(−23.5−40.7i)T2 1 + (0.133 - 0.232i)T + (-23.5 - 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(7.19+12.4i)T+(−29.5+51.0i)T2 1 + (7.19 + 12.4i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.23+7.33i)T+(−30.5−52.8i)T2 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.13−5.42i)T+(−33.5+58.0i)T2 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2}
71 1−9.46T+71T2 1 - 9.46T + 71T^{2}
73 1−6.92T+73T2 1 - 6.92T + 73T^{2}
79 1+(−7.73+13.3i)T+(−39.5−68.4i)T2 1 + (-7.73 + 13.3i)T + (-39.5 - 68.4i)T^{2}
83 1+(−6.59+11.4i)T+(−41.5−71.8i)T2 1 + (-6.59 + 11.4i)T + (-41.5 - 71.8i)T^{2}
89 1−9.92T+89T2 1 - 9.92T + 89T^{2}
97 1+(−4.46+7.73i)T+(−48.5−84.0i)T2 1 + (-4.46 + 7.73i)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

−9.249747887127768477532381543510, −8.178335393956246074485229823980, −7.72677965354023007688876812490, −6.43680580679924451104690349274, −6.19551012861373935479170853890, −4.85742516698642617113826521396, −4.25053322852681914746467729030, −3.33488503960487398422299693220, −2.06816402390783266582383411502, −0.891179820985886843111427235684, 0.907493022977264738873338658089, 2.30373378258580713665776380240, 3.33876197036055713726498017737, 4.05981116256698429181254831358, 5.24528185166705126556254737932, 5.88774781281098208727645206792, 6.80724067753730119773254501682, 7.62979002780030014816723831612, 8.193996777730695381301405319170, 9.120575416160271930426646438211

Graph of the ZZ-function along the critical line