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Properties

Label 2-170-85.62-c1-0-4
Degree 22
Conductor 170170
Sign 0.984−0.172i0.984 - 0.172i
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  + (0.382 + 0.923i)2-s + (−1.16 − 0.776i)3-s + (−0.707 + 0.707i)4-s + (1.31 − 1.80i)5-s + (0.272 − 1.37i)6-s + (3.77 + 0.749i)7-s + (−0.923 − 0.382i)8-s + (−0.399 − 0.965i)9-s + (2.17 + 0.524i)10-s + (3.05 + 0.606i)11-s + (1.37 − 0.272i)12-s + 4.47i·13-s + (0.749 + 3.77i)14-s + (−2.93 + 1.07i)15-s − i·16-s + (−3.21 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.671 − 0.448i)3-s + (−0.353 + 0.353i)4-s + (0.588 − 0.808i)5-s + (0.111 − 0.559i)6-s + (1.42 + 0.283i)7-s + (−0.326 − 0.135i)8-s + (−0.133 − 0.321i)9-s + (0.687 + 0.165i)10-s + (0.919 + 0.182i)11-s + (0.395 − 0.0787i)12-s + 1.24i·13-s + (0.200 + 1.00i)14-s + (−0.757 + 0.278i)15-s − 0.250i·16-s + (−0.780 − 0.625i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.984−0.172i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.984−0.172i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.984−0.172i0.984 - 0.172i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(147,⋅)\chi_{170} (147, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.984−0.172i)(2,\ 170,\ (\ :1/2),\ 0.984 - 0.172i)

Particular Values

L(1)L(1) ≈\approx 1.24484+0.108253i1.24484 + 0.108253i
L(12)L(\frac12) ≈\approx 1.24484+0.108253i1.24484 + 0.108253i
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+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
5 1+(−1.31+1.80i)T 1 + (-1.31 + 1.80i)T
17 1+(3.21+2.57i)T 1 + (3.21 + 2.57i)T
good3 1+(1.16+0.776i)T+(1.14+2.77i)T2 1 + (1.16 + 0.776i)T + (1.14 + 2.77i)T^{2}
7 1+(−3.77−0.749i)T+(6.46+2.67i)T2 1 + (-3.77 - 0.749i)T + (6.46 + 2.67i)T^{2}
11 1+(−3.05−0.606i)T+(10.1+4.20i)T2 1 + (-3.05 - 0.606i)T + (10.1 + 4.20i)T^{2}
13 1−4.47iT−13T2 1 - 4.47iT - 13T^{2}
19 1+(−1.25+3.02i)T+(−13.4−13.4i)T2 1 + (-1.25 + 3.02i)T + (-13.4 - 13.4i)T^{2}
23 1+(−1.36−2.05i)T+(−8.80+21.2i)T2 1 + (-1.36 - 2.05i)T + (-8.80 + 21.2i)T^{2}
29 1+(5.13+3.42i)T+(11.0+26.7i)T2 1 + (5.13 + 3.42i)T + (11.0 + 26.7i)T^{2}
31 1+(6.04−1.20i)T+(28.6−11.8i)T2 1 + (6.04 - 1.20i)T + (28.6 - 11.8i)T^{2}
37 1+(4.21−6.31i)T+(−14.1−34.1i)T2 1 + (4.21 - 6.31i)T + (-14.1 - 34.1i)T^{2}
41 1+(−1.67+1.12i)T+(15.6−37.8i)T2 1 + (-1.67 + 1.12i)T + (15.6 - 37.8i)T^{2}
43 1+(2.41−5.82i)T+(−30.4−30.4i)T2 1 + (2.41 - 5.82i)T + (-30.4 - 30.4i)T^{2}
47 1−9.31T+47T2 1 - 9.31T + 47T^{2}
53 1+(11.5−4.76i)T+(37.4−37.4i)T2 1 + (11.5 - 4.76i)T + (37.4 - 37.4i)T^{2}
59 1+(−4.66+1.93i)T+(41.7−41.7i)T2 1 + (-4.66 + 1.93i)T + (41.7 - 41.7i)T^{2}
61 1+(1.54+2.31i)T+(−23.3+56.3i)T2 1 + (1.54 + 2.31i)T + (-23.3 + 56.3i)T^{2}
67 1+(6.06−6.06i)T−67iT2 1 + (6.06 - 6.06i)T - 67iT^{2}
71 1+(−0.137−0.691i)T+(−65.5+27.1i)T2 1 + (-0.137 - 0.691i)T + (-65.5 + 27.1i)T^{2}
73 1+(−1.32+0.264i)T+(67.4−27.9i)T2 1 + (-1.32 + 0.264i)T + (67.4 - 27.9i)T^{2}
79 1+(2.06−10.3i)T+(−72.9−30.2i)T2 1 + (2.06 - 10.3i)T + (-72.9 - 30.2i)T^{2}
83 1+(5.12+12.3i)T+(−58.6+58.6i)T2 1 + (5.12 + 12.3i)T + (-58.6 + 58.6i)T^{2}
89 1+(−4.33+4.33i)T−89iT2 1 + (-4.33 + 4.33i)T - 89iT^{2}
97 1+(−7.72+1.53i)T+(89.6−37.1i)T2 1 + (-7.72 + 1.53i)T + (89.6 - 37.1i)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

−12.82385712665609443874991239534, −11.69888599133680337699161212399, −11.40827613200954908984374391209, −9.276455225601302613081156052461, −8.861397961493905142803514296169, −7.36011873737450341946109705452, −6.36415121534161056412341759335, −5.27171219587902343019615333107, −4.40036726873754081214286906879, −1.61899804794579686914599747989, 1.90874059989789529884476927391, 3.75870799482099279170197064470, 5.12143485841678369751492796065, 5.94586182563819022704577205493, 7.54250215492414513264874988622, 8.869860591433497376091453437374, 10.28129154042894722854322768937, 10.89030659917989512114087161783, 11.33777290991688866196263166533, 12.61460458820042375069386845277

Graph of the ZZ-function along the critical line