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Properties

Label 2-170-17.16-c1-0-5
Degree 22
Conductor 170170
Sign 0.242+0.970i0.242 + 0.970i
Analytic cond. 1.357451.35745
Root an. cond. 1.165091.16509
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  + 2-s − 3i·3-s + 4-s − i·5-s − 3i·6-s + 4i·7-s + 8-s − 6·9-s − i·10-s − 2i·11-s − 3i·12-s + 13-s + 4i·14-s − 3·15-s + 16-s + (−4 + i)17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73i·3-s + 0.5·4-s − 0.447i·5-s − 1.22i·6-s + 1.51i·7-s + 0.353·8-s − 2·9-s − 0.316i·10-s − 0.603i·11-s − 0.866i·12-s + 0.277·13-s + 1.06i·14-s − 0.774·15-s + 0.250·16-s + (−0.970 + 0.242i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.242+0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.242+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.242+0.970i0.242 + 0.970i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(101,⋅)\chi_{170} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.242+0.970i)(2,\ 170,\ (\ :1/2),\ 0.242 + 0.970i)

Particular Values

L(1)L(1) ≈\approx 1.29796−1.01342i1.29796 - 1.01342i
L(12)L(\frac12) ≈\approx 1.29796−1.01342i1.29796 - 1.01342i
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−T 1 - T
5 1+iT 1 + iT
17 1+(4−i)T 1 + (4 - i)T
good3 1+3iT−3T2 1 + 3iT - 3T^{2}
7 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
19 1−7T+19T2 1 - 7T + 19T^{2}
23 1−6iT−23T2 1 - 6iT - 23T^{2}
29 1+3iT−29T2 1 + 3iT - 29T^{2}
31 1−7iT−31T2 1 - 7iT - 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+8iT−41T2 1 + 8iT - 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1−9T+47T2 1 - 9T + 47T^{2}
53 1+11T+53T2 1 + 11T + 53T^{2}
59 1+5T+59T2 1 + 5T + 59T^{2}
61 1−iT−61T2 1 - iT - 61T^{2}
67 1+10T+67T2 1 + 10T + 67T^{2}
71 1+iT−71T2 1 + iT - 71T^{2}
73 1−9iT−73T2 1 - 9iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+T+89T2 1 + T + 89T^{2}
97 1+iT−97T2 1 + iT - 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

−12.51617759768662949677062718120, −11.96265363250534614982759832729, −11.25111737501018010894859849339, −9.149973819366366417744698990508, −8.290761348738417328071218177964, −7.18802883405412538593069410516, −6.01869006498360600313377989302, −5.37230424282602979695769710811, −3.04581874417342468555685221023, −1.70330108837015106579750316817, 3.14050325071886133800571108651, 4.19873050916616556771392415588, 4.89538188592078517366076686232, 6.46767238726875828182751799868, 7.68616206475835792833706874124, 9.333687644170166982705742887466, 10.23045900052884807800347481520, 10.84344936356113208565349317874, 11.67626958654991442426548564908, 13.30155021929119161366332252046

Graph of the ZZ-function along the critical line