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Properties

Label 2-1085-7.2-c1-0-73
Degree 22
Conductor 10851085
Sign −0.705−0.708i-0.705 - 0.708i
Analytic cond. 8.663768.66376
Root an. cond. 2.943422.94342
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  + (0.773 − 1.34i)2-s + (−1.14 − 1.97i)3-s + (−0.197 − 0.342i)4-s + (0.5 − 0.866i)5-s − 3.52·6-s + (−2.50 − 0.851i)7-s + 2.48·8-s + (−1.09 + 1.90i)9-s + (−0.773 − 1.34i)10-s + (−2.75 − 4.76i)11-s + (−0.451 + 0.781i)12-s + 3.50·13-s + (−3.08 + 2.69i)14-s − 2.28·15-s + (2.31 − 4.01i)16-s + (−1.13 − 1.95i)17-s + ⋯
L(s)  = 1  + (0.547 − 0.947i)2-s + (−0.658 − 1.14i)3-s + (−0.0989 − 0.171i)4-s + (0.223 − 0.387i)5-s − 1.44·6-s + (−0.946 − 0.321i)7-s + 0.877·8-s + (−0.366 + 0.634i)9-s + (−0.244 − 0.423i)10-s + (−0.830 − 1.43i)11-s + (−0.130 + 0.225i)12-s + 0.970·13-s + (−0.823 + 0.721i)14-s − 0.588·15-s + (0.579 − 1.00i)16-s + (−0.274 − 0.475i)17-s + ⋯

Functional equation

Λ(s)=(1085s/2 ΓC(s) L(s)=((−0.705−0.708i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1085s/2 ΓC(s+1/2) L(s)=((−0.705−0.708i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10851085    =    5⋅7⋅315 \cdot 7 \cdot 31
Sign: −0.705−0.708i-0.705 - 0.708i
Analytic conductor: 8.663768.66376
Root analytic conductor: 2.943422.94342
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1085(156,⋅)\chi_{1085} (156, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1085, ( :1/2), −0.705−0.708i)(2,\ 1085,\ (\ :1/2),\ -0.705 - 0.708i)

Particular Values

L(1)L(1) ≈\approx 1.3610825711.361082571
L(12)L(\frac12) ≈\approx 1.3610825711.361082571
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)
bad5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(2.50+0.851i)T 1 + (2.50 + 0.851i)T
31 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(−0.773+1.34i)T+(−1−1.73i)T2 1 + (-0.773 + 1.34i)T + (-1 - 1.73i)T^{2}
3 1+(1.14+1.97i)T+(−1.5+2.59i)T2 1 + (1.14 + 1.97i)T + (-1.5 + 2.59i)T^{2}
11 1+(2.75+4.76i)T+(−5.5+9.52i)T2 1 + (2.75 + 4.76i)T + (-5.5 + 9.52i)T^{2}
13 1−3.50T+13T2 1 - 3.50T + 13T^{2}
17 1+(1.13+1.95i)T+(−8.5+14.7i)T2 1 + (1.13 + 1.95i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.329+0.570i)T+(−9.5−16.4i)T2 1 + (-0.329 + 0.570i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.74−4.75i)T+(−11.5−19.9i)T2 1 + (2.74 - 4.75i)T + (-11.5 - 19.9i)T^{2}
29 1−1.83T+29T2 1 - 1.83T + 29T^{2}
37 1+(4.69−8.13i)T+(−18.5−32.0i)T2 1 + (4.69 - 8.13i)T + (-18.5 - 32.0i)T^{2}
41 1−6.10T+41T2 1 - 6.10T + 41T^{2}
43 1+7.49T+43T2 1 + 7.49T + 43T^{2}
47 1+(1.60−2.78i)T+(−23.5−40.7i)T2 1 + (1.60 - 2.78i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.0757+0.131i)T+(−26.5+45.8i)T2 1 + (0.0757 + 0.131i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.64+9.78i)T+(−29.5+51.0i)T2 1 + (5.64 + 9.78i)T + (-29.5 + 51.0i)T^{2}
61 1+(−7.23+12.5i)T+(−30.5−52.8i)T2 1 + (-7.23 + 12.5i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.23−10.8i)T+(−33.5+58.0i)T2 1 + (-6.23 - 10.8i)T + (-33.5 + 58.0i)T^{2}
71 1+6.50T+71T2 1 + 6.50T + 71T^{2}
73 1+(6.77+11.7i)T+(−36.5+63.2i)T2 1 + (6.77 + 11.7i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.48−6.04i)T+(−39.5−68.4i)T2 1 + (3.48 - 6.04i)T + (-39.5 - 68.4i)T^{2}
83 1−16.8T+83T2 1 - 16.8T + 83T^{2}
89 1+(−4.74+8.21i)T+(−44.5−77.0i)T2 1 + (-4.74 + 8.21i)T + (-44.5 - 77.0i)T^{2}
97 1−6.36T+97T2 1 - 6.36T + 97T^{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

−9.593522471511019438270995212126, −8.393690335938217211405781588756, −7.64955378724450786321928597101, −6.63367498126924856782736455645, −5.97745033108578794843163881587, −5.07740472610326529959720961301, −3.68378856843137760764134956801, −2.97196140229120209993465748372, −1.63058723794915413283151212389, −0.53979036689454361610134991040, 2.19626658921603389802311233423, 3.74475993389294989353189532703, 4.50707094721039326057332674930, 5.38319415458406046769468087880, 6.03604294272789617597868993257, 6.73372401665974894084009919174, 7.61055213168473431681305576979, 8.798962050581249029942285118068, 9.830924378993140321154486140455, 10.43560727122880083392871701392

Graph of the ZZ-function along the critical line