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Properties

Label 2-2268-21.17-c1-0-3
Degree 22
Conductor 22682268
Sign 0.315−0.948i0.315 - 0.948i
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  + (−1.48 − 2.57i)5-s + (−2.18 + 1.49i)7-s + (4.09 + 2.36i)11-s − 4.08i·13-s + (−0.835 + 1.44i)17-s + (−4.25 + 2.45i)19-s + (−4.25 + 2.45i)23-s + (−1.91 + 3.30i)25-s − 0.275i·29-s + (1.38 + 0.801i)31-s + (7.08 + 3.40i)35-s + (−1.69 − 2.93i)37-s + 7.11·41-s − 10.4·43-s + (5.49 + 9.52i)47-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)5-s + (−0.825 + 0.564i)7-s + (1.23 + 0.712i)11-s − 1.13i·13-s + (−0.202 + 0.350i)17-s + (−0.975 + 0.563i)19-s + (−0.886 + 0.511i)23-s + (−0.382 + 0.661i)25-s − 0.0511i·29-s + (0.249 + 0.143i)31-s + (1.19 + 0.575i)35-s + (−0.278 − 0.483i)37-s + 1.11·41-s − 1.59·43-s + (0.802 + 1.38i)47-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.315−0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.315−0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.315−0.948i0.315 - 0.948i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(1781,⋅)\chi_{2268} (1781, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.315−0.948i)(2,\ 2268,\ (\ :1/2),\ 0.315 - 0.948i)

Particular Values

L(1)L(1) ≈\approx 0.87622566330.8762256633
L(12)L(\frac12) ≈\approx 0.87622566330.8762256633
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
7 1+(2.18−1.49i)T 1 + (2.18 - 1.49i)T
good5 1+(1.48+2.57i)T+(−2.5+4.33i)T2 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2}
11 1+(−4.09−2.36i)T+(5.5+9.52i)T2 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2}
13 1+4.08iT−13T2 1 + 4.08iT - 13T^{2}
17 1+(0.835−1.44i)T+(−8.5−14.7i)T2 1 + (0.835 - 1.44i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.25−2.45i)T+(9.5−16.4i)T2 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2}
23 1+(4.25−2.45i)T+(11.5−19.9i)T2 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2}
29 1+0.275iT−29T2 1 + 0.275iT - 29T^{2}
31 1+(−1.38−0.801i)T+(15.5+26.8i)T2 1 + (-1.38 - 0.801i)T + (15.5 + 26.8i)T^{2}
37 1+(1.69+2.93i)T+(−18.5+32.0i)T2 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2}
41 1−7.11T+41T2 1 - 7.11T + 41T^{2}
43 1+10.4T+43T2 1 + 10.4T + 43T^{2}
47 1+(−5.49−9.52i)T+(−23.5+40.7i)T2 1 + (-5.49 - 9.52i)T + (-23.5 + 40.7i)T^{2}
53 1+(−0.707−0.408i)T+(26.5+45.8i)T2 1 + (-0.707 - 0.408i)T + (26.5 + 45.8i)T^{2}
59 1+(1.37−2.38i)T+(−29.5−51.0i)T2 1 + (1.37 - 2.38i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6.23+3.60i)T+(30.5−52.8i)T2 1 + (-6.23 + 3.60i)T + (30.5 - 52.8i)T^{2}
67 1+(5.80−10.0i)T+(−33.5−58.0i)T2 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2}
71 1−10.4iT−71T2 1 - 10.4iT - 71T^{2}
73 1+(−13.6−7.88i)T+(36.5+63.2i)T2 1 + (-13.6 - 7.88i)T + (36.5 + 63.2i)T^{2}
79 1+(−6.15−10.6i)T+(−39.5+68.4i)T2 1 + (-6.15 - 10.6i)T + (-39.5 + 68.4i)T^{2}
83 1−8.07T+83T2 1 - 8.07T + 83T^{2}
89 1+(−4.60−7.98i)T+(−44.5+77.0i)T2 1 + (-4.60 - 7.98i)T + (-44.5 + 77.0i)T^{2}
97 1+8.09iT−97T2 1 + 8.09iT - 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

−9.133782055787798233146265521909, −8.414893077933915314575364665738, −7.85957996816750108791044095352, −6.77440520955830855802071679773, −6.03080862833854464530320713127, −5.23178375574838875865903117738, −4.16638791506610097211341621984, −3.71118721151742891399681893617, −2.33603021366660940710563189366, −1.07169189452388431478348008171, 0.34740718249722135407090889959, 2.07203908360582233476087489028, 3.24754487980293897243575095962, 3.83987898616740940586233413824, 4.56521522862857567614180594669, 6.17870381820314971027510652602, 6.62496685221450605689918244206, 7.03254517519833082084195980982, 8.057174571304897117444918510698, 8.957228412632442948969772383752

Graph of the ZZ-function along the critical line