Properties

Label 2-1170-13.12-c1-0-23
Degree 22
Conductor 11701170
Sign 1-1
Analytic cond. 9.342499.34249
Root an. cond. 3.056543.05654
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  i·2-s − 4-s i·5-s − 4.60i·7-s + i·8-s − 10-s + 3.60·13-s − 4.60·14-s + 16-s − 4.60·17-s − 4.60i·19-s + i·20-s + 1.39·23-s − 25-s − 3.60i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 1.74i·7-s + 0.353i·8-s − 0.316·10-s + 1.00·13-s − 1.23·14-s + 0.250·16-s − 1.11·17-s − 1.05i·19-s + 0.223i·20-s + 0.290·23-s − 0.200·25-s − 0.707i·26-s + ⋯

Functional equation

Λ(s)=(1170s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1170s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11701170    =    2325132 \cdot 3^{2} \cdot 5 \cdot 13
Sign: 1-1
Analytic conductor: 9.342499.34249
Root analytic conductor: 3.056543.05654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1170(181,)\chi_{1170} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1170, ( :1/2), 1)(2,\ 1170,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) \approx 1.1700070641.170007064
L(12)L(\frac12) \approx 1.1700070641.170007064
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+iT 1 + iT
3 1 1
5 1+iT 1 + iT
13 13.60T 1 - 3.60T
good7 1+4.60iT7T2 1 + 4.60iT - 7T^{2}
11 111T2 1 - 11T^{2}
17 1+4.60T+17T2 1 + 4.60T + 17T^{2}
19 1+4.60iT19T2 1 + 4.60iT - 19T^{2}
23 11.39T+23T2 1 - 1.39T + 23T^{2}
29 1+4.60T+29T2 1 + 4.60T + 29T^{2}
31 16iT31T2 1 - 6iT - 31T^{2}
37 1+9.21iT37T2 1 + 9.21iT - 37T^{2}
41 1+3.21iT41T2 1 + 3.21iT - 41T^{2}
43 18T+43T2 1 - 8T + 43T^{2}
47 19.21iT47T2 1 - 9.21iT - 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 19.21iT59T2 1 - 9.21iT - 59T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1+3.21iT67T2 1 + 3.21iT - 67T^{2}
71 1+9.21iT71T2 1 + 9.21iT - 71T^{2}
73 1+1.39iT73T2 1 + 1.39iT - 73T^{2}
79 1+14.4T+79T2 1 + 14.4T + 79T^{2}
83 1+2.78iT83T2 1 + 2.78iT - 83T^{2}
89 1+15.2iT89T2 1 + 15.2iT - 89T^{2}
97 11.39iT97T2 1 - 1.39iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.252645784952432429332199795045, −8.891767460786387990176091042533, −7.69396526152934103147073935464, −7.03274628397051416972650465303, −5.95308081148676960619149608294, −4.63297653432515527542396250964, −4.16042112515793033810378920628, −3.14775777238510614307352411497, −1.62545250391783214636567988570, −0.51487114833764607901905081384, 1.90099776105065766427192433005, 3.06447311149039710998548026074, 4.21200496971249717896478680118, 5.40331199792332387568937544010, 6.04304329254448087229724277265, 6.64149625038635540495788379015, 7.85416547768879583175197780591, 8.525544310527710369167190329911, 9.161138011573199076177604006441, 9.945491203669175515547786614445

Graph of the ZZ-function along the critical line