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Properties

Label 2-170-17.2-c1-0-4
Degree 22
Conductor 170170
Sign 0.890−0.454i0.890 - 0.454i
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  + (0.707 + 0.707i)2-s + (1.36 − 0.564i)3-s + 1.00i·4-s + (0.382 + 0.923i)5-s + (1.36 + 0.564i)6-s + (1.35 − 3.27i)7-s + (−0.707 + 0.707i)8-s + (−0.581 + 0.581i)9-s + (−0.382 + 0.923i)10-s + (−4.35 − 1.80i)11-s + (0.564 + 1.36i)12-s + 5.47i·13-s + (3.27 − 1.35i)14-s + (1.04 + 1.04i)15-s − 1.00·16-s + (2.52 − 3.25i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.787 − 0.326i)3-s + 0.500i·4-s + (0.171 + 0.413i)5-s + (0.556 + 0.230i)6-s + (0.513 − 1.23i)7-s + (−0.250 + 0.250i)8-s + (−0.193 + 0.193i)9-s + (−0.121 + 0.292i)10-s + (−1.31 − 0.543i)11-s + (0.163 + 0.393i)12-s + 1.51i·13-s + (0.876 − 0.362i)14-s + (0.269 + 0.269i)15-s − 0.250·16-s + (0.613 − 0.789i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.890−0.454i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.890−0.454i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.890−0.454i0.890 - 0.454i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(121,⋅)\chi_{170} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.890−0.454i)(2,\ 170,\ (\ :1/2),\ 0.890 - 0.454i)

Particular Values

L(1)L(1) ≈\approx 1.72813+0.414910i1.72813 + 0.414910i
L(12)L(\frac12) ≈\approx 1.72813+0.414910i1.72813 + 0.414910i
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
17 1+(−2.52+3.25i)T 1 + (-2.52 + 3.25i)T
good3 1+(−1.36+0.564i)T+(2.12−2.12i)T2 1 + (-1.36 + 0.564i)T + (2.12 - 2.12i)T^{2}
7 1+(−1.35+3.27i)T+(−4.94−4.94i)T2 1 + (-1.35 + 3.27i)T + (-4.94 - 4.94i)T^{2}
11 1+(4.35+1.80i)T+(7.77+7.77i)T2 1 + (4.35 + 1.80i)T + (7.77 + 7.77i)T^{2}
13 1−5.47iT−13T2 1 - 5.47iT - 13T^{2}
19 1+(−0.857−0.857i)T+19iT2 1 + (-0.857 - 0.857i)T + 19iT^{2}
23 1+(5.28+2.18i)T+(16.2+16.2i)T2 1 + (5.28 + 2.18i)T + (16.2 + 16.2i)T^{2}
29 1+(1.77+4.29i)T+(−20.5+20.5i)T2 1 + (1.77 + 4.29i)T + (-20.5 + 20.5i)T^{2}
31 1+(−6.72+2.78i)T+(21.9−21.9i)T2 1 + (-6.72 + 2.78i)T + (21.9 - 21.9i)T^{2}
37 1+(8.58−3.55i)T+(26.1−26.1i)T2 1 + (8.58 - 3.55i)T + (26.1 - 26.1i)T^{2}
41 1+(−0.0547+0.132i)T+(−28.9−28.9i)T2 1 + (-0.0547 + 0.132i)T + (-28.9 - 28.9i)T^{2}
43 1+(0.587−0.587i)T−43iT2 1 + (0.587 - 0.587i)T - 43iT^{2}
47 1+5.85iT−47T2 1 + 5.85iT - 47T^{2}
53 1+(−6.54−6.54i)T+53iT2 1 + (-6.54 - 6.54i)T + 53iT^{2}
59 1+(3.33−3.33i)T−59iT2 1 + (3.33 - 3.33i)T - 59iT^{2}
61 1+(−4.76+11.4i)T+(−43.1−43.1i)T2 1 + (-4.76 + 11.4i)T + (-43.1 - 43.1i)T^{2}
67 1−8.66T+67T2 1 - 8.66T + 67T^{2}
71 1+(−4.75+1.96i)T+(50.2−50.2i)T2 1 + (-4.75 + 1.96i)T + (50.2 - 50.2i)T^{2}
73 1+(−6.17−14.9i)T+(−51.6+51.6i)T2 1 + (-6.17 - 14.9i)T + (-51.6 + 51.6i)T^{2}
79 1+(−9.34−3.87i)T+(55.8+55.8i)T2 1 + (-9.34 - 3.87i)T + (55.8 + 55.8i)T^{2}
83 1+(−2.40−2.40i)T+83iT2 1 + (-2.40 - 2.40i)T + 83iT^{2}
89 1−2.24iT−89T2 1 - 2.24iT - 89T^{2}
97 1+(−1.66−4.02i)T+(−68.5+68.5i)T2 1 + (-1.66 - 4.02i)T + (-68.5 + 68.5i)T^{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

−13.48527967756984319563869752967, −11.93340475843217307734455738669, −10.92717095718889013636888118660, −9.826410352954333617696921078013, −8.299384093439368943093016698727, −7.69476429607915333021088938358, −6.71607394649876030504279754078, −5.21445117041883226244285705797, −3.86255822775029665986653948268, −2.40532095765347409643954785236, 2.26074361973846656271127409516, 3.37017107446711173572010687131, 5.10940936921285466603210446442, 5.75362912733269015170707705360, 7.937539710473713299507249241118, 8.595257446648270767966260020264, 9.794036225432959731058348655474, 10.58944734047855909604616509939, 12.05237874406556887305949261434, 12.58333474375168429035590086074

Graph of the ZZ-function along the critical line