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Properties

Label 2-84-28.27-c1-0-3
Degree 22
Conductor 8484
Sign 0.997+0.0636i0.997 + 0.0636i
Analytic cond. 0.6707430.670743
Root an. cond. 0.8189890.818989
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  + (−1.28 + 0.599i)2-s + 3-s + (1.28 − 1.53i)4-s − 3.33i·5-s + (−1.28 + 0.599i)6-s + (1.56 + 2.13i)7-s + (−0.719 + 2.73i)8-s + 9-s + (2 + 4.27i)10-s + 0.936i·11-s + (1.28 − 1.53i)12-s − 1.87i·13-s + (−3.28 − 1.79i)14-s − 3.33i·15-s + (−0.719 − 3.93i)16-s + 5.20i·17-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + 0.577·3-s + (0.640 − 0.768i)4-s − 1.49i·5-s + (−0.522 + 0.244i)6-s + (0.590 + 0.807i)7-s + (−0.254 + 0.967i)8-s + 0.333·9-s + (0.632 + 1.35i)10-s + 0.282i·11-s + (0.369 − 0.443i)12-s − 0.519i·13-s + (−0.876 − 0.480i)14-s − 0.861i·15-s + (−0.179 − 0.983i)16-s + 1.26i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.997+0.0636i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+1/2) L(s)=((0.997+0.0636i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.997+0.0636i0.997 + 0.0636i
Analytic conductor: 0.6707430.670743
Root analytic conductor: 0.8189890.818989
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ84(55,⋅)\chi_{84} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :1/2), 0.997+0.0636i)(2,\ 84,\ (\ :1/2),\ 0.997 + 0.0636i)

Particular Values

L(1)L(1) ≈\approx 0.802807−0.0255711i0.802807 - 0.0255711i
L(12)L(\frac12) ≈\approx 0.802807−0.0255711i0.802807 - 0.0255711i
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+(1.28−0.599i)T 1 + (1.28 - 0.599i)T
3 1−T 1 - T
7 1+(−1.56−2.13i)T 1 + (-1.56 - 2.13i)T
good5 1+3.33iT−5T2 1 + 3.33iT - 5T^{2}
11 1−0.936iT−11T2 1 - 0.936iT - 11T^{2}
13 1+1.87iT−13T2 1 + 1.87iT - 13T^{2}
17 1−5.20iT−17T2 1 - 5.20iT - 17T^{2}
19 1+7.12T+19T2 1 + 7.12T + 19T^{2}
23 1+0.936iT−23T2 1 + 0.936iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−1.12T+37T2 1 - 1.12T + 37T^{2}
41 1−1.46iT−41T2 1 - 1.46iT - 41T^{2}
43 1−9.06iT−43T2 1 - 9.06iT - 43T^{2}
47 1+6.24T+47T2 1 + 6.24T + 47T^{2}
53 1−12.2T+53T2 1 - 12.2T + 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+4.79iT−61T2 1 + 4.79iT - 61T^{2}
67 1+10.9iT−67T2 1 + 10.9iT - 67T^{2}
71 1+3.86iT−71T2 1 + 3.86iT - 71T^{2}
73 1+6.67iT−73T2 1 + 6.67iT - 73T^{2}
79 1+2.39iT−79T2 1 + 2.39iT - 79T^{2}
83 1−10.2T+83T2 1 - 10.2T + 83T^{2}
89 1+1.46iT−89T2 1 + 1.46iT - 89T^{2}
97 1−10.4iT−97T2 1 - 10.4iT - 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

−14.76819994238536580665720511066, −13.07056018224511956699937647030, −12.22863885781769276542438268423, −10.75320245503232241960206349880, −9.417286413170718152697370458412, −8.510815628725766047549570632269, −8.036450704565657382217270384451, −6.08336522055488476693531156665, −4.73027934549930739366738681298, −1.85079309095007750368930363162, 2.38992235256116385514090455262, 3.85746371001894976005028043319, 6.74425116675161729432485668747, 7.45663452075054569364412035995, 8.718749724391334480388567760508, 10.06560209788034286652920546956, 10.84036931716501584657900716808, 11.67699098899141575774887647947, 13.36373791091753852187686087932, 14.34846655778372350742695271801

Graph of the ZZ-function along the critical line