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Properties

Label 2-170-17.15-c1-0-2
Degree 22
Conductor 170170
Sign 0.885+0.465i0.885 + 0.465i
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 + (−0.0294 − 0.0709i)3-s + 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.0294 + 0.0709i)6-s + (3.48 + 1.44i)7-s + (0.707 − 0.707i)8-s + (2.11 − 2.11i)9-s + (0.923 + 0.382i)10-s + (1.39 − 3.37i)11-s + (0.0709 − 0.0294i)12-s + 4.29i·13-s + (−1.44 − 3.48i)14-s + (0.0543 + 0.0543i)15-s − 1.00·16-s + (2.96 − 2.86i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.0169 − 0.0409i)3-s + 0.500i·4-s + (−0.413 + 0.171i)5-s + (−0.0120 + 0.0289i)6-s + (1.31 + 0.546i)7-s + (0.250 − 0.250i)8-s + (0.705 − 0.705i)9-s + (0.292 + 0.121i)10-s + (0.421 − 1.01i)11-s + (0.0204 − 0.00848i)12-s + 1.19i·13-s + (−0.386 − 0.932i)14-s + (0.0140 + 0.0140i)15-s − 0.250·16-s + (0.718 − 0.696i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.885+0.465i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.885+0.465i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.885+0.465i0.885 + 0.465i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(151,⋅)\chi_{170} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.885+0.465i)(2,\ 170,\ (\ :1/2),\ 0.885 + 0.465i)

Particular Values

L(1)L(1) ≈\approx 0.962118−0.237592i0.962118 - 0.237592i
L(12)L(\frac12) ≈\approx 0.962118−0.237592i0.962118 - 0.237592i
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+(0.923−0.382i)T 1 + (0.923 - 0.382i)T
17 1+(−2.96+2.86i)T 1 + (-2.96 + 2.86i)T
good3 1+(0.0294+0.0709i)T+(−2.12+2.12i)T2 1 + (0.0294 + 0.0709i)T + (-2.12 + 2.12i)T^{2}
7 1+(−3.48−1.44i)T+(4.94+4.94i)T2 1 + (-3.48 - 1.44i)T + (4.94 + 4.94i)T^{2}
11 1+(−1.39+3.37i)T+(−7.77−7.77i)T2 1 + (-1.39 + 3.37i)T + (-7.77 - 7.77i)T^{2}
13 1−4.29iT−13T2 1 - 4.29iT - 13T^{2}
19 1+(2.18+2.18i)T+19iT2 1 + (2.18 + 2.18i)T + 19iT^{2}
23 1+(2.47−5.98i)T+(−16.2−16.2i)T2 1 + (2.47 - 5.98i)T + (-16.2 - 16.2i)T^{2}
29 1+(−2.10+0.871i)T+(20.5−20.5i)T2 1 + (-2.10 + 0.871i)T + (20.5 - 20.5i)T^{2}
31 1+(−2.31−5.58i)T+(−21.9+21.9i)T2 1 + (-2.31 - 5.58i)T + (-21.9 + 21.9i)T^{2}
37 1+(3.10+7.49i)T+(−26.1+26.1i)T2 1 + (3.10 + 7.49i)T + (-26.1 + 26.1i)T^{2}
41 1+(7.61+3.15i)T+(28.9+28.9i)T2 1 + (7.61 + 3.15i)T + (28.9 + 28.9i)T^{2}
43 1+(7.80−7.80i)T−43iT2 1 + (7.80 - 7.80i)T - 43iT^{2}
47 1−8.27iT−47T2 1 - 8.27iT - 47T^{2}
53 1+(7.36+7.36i)T+53iT2 1 + (7.36 + 7.36i)T + 53iT^{2}
59 1+(3.39−3.39i)T−59iT2 1 + (3.39 - 3.39i)T - 59iT^{2}
61 1+(2.56+1.06i)T+(43.1+43.1i)T2 1 + (2.56 + 1.06i)T + (43.1 + 43.1i)T^{2}
67 1−3.43T+67T2 1 - 3.43T + 67T^{2}
71 1+(5.25+12.6i)T+(−50.2+50.2i)T2 1 + (5.25 + 12.6i)T + (-50.2 + 50.2i)T^{2}
73 1+(−10.9+4.53i)T+(51.6−51.6i)T2 1 + (-10.9 + 4.53i)T + (51.6 - 51.6i)T^{2}
79 1+(−1.15+2.80i)T+(−55.8−55.8i)T2 1 + (-1.15 + 2.80i)T + (-55.8 - 55.8i)T^{2}
83 1+(1.30+1.30i)T+83iT2 1 + (1.30 + 1.30i)T + 83iT^{2}
89 1−9.64iT−89T2 1 - 9.64iT - 89T^{2}
97 1+(0.613−0.254i)T+(68.5−68.5i)T2 1 + (0.613 - 0.254i)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

−12.18869462853600879718178528536, −11.68755064872605309689219219163, −10.95262382147450411212884691876, −9.557770404544890043202703891275, −8.725725171995849662238897941454, −7.72775018042051086988705359343, −6.52943823059306203441378910511, −4.82610255980682381290886194147, −3.48789258375340691781226307507, −1.57124505758041150929784473835, 1.62336883238961192903559128684, 4.24336063277552342083638138420, 5.15303798970325904065632917820, 6.80368706150279002961889988224, 7.996518555374194391316048728988, 8.227118791302506665496513562051, 10.14845623117433491917440763084, 10.45023426969981828487711011347, 11.81350673300795919821650299180, 12.81444158509702673141441832751

Graph of the ZZ-function along the critical line