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Properties

Label 2-85-17.2-c1-0-4
Degree 22
Conductor 8585
Sign 0.294+0.955i0.294 + 0.955i
Analytic cond. 0.6787280.678728
Root an. cond. 0.8238490.823849
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  + (−1.09 − 1.09i)2-s + (2.77 − 1.15i)3-s + 0.419i·4-s + (0.382 + 0.923i)5-s + (−4.32 − 1.79i)6-s + (−1.32 + 3.19i)7-s + (−1.73 + 1.73i)8-s + (4.27 − 4.27i)9-s + (0.595 − 1.43i)10-s + (−3.92 − 1.62i)11-s + (0.483 + 1.16i)12-s + 0.127i·13-s + (4.96 − 2.05i)14-s + (2.12 + 2.12i)15-s + 4.66·16-s + (−4.11 − 0.193i)17-s + ⋯
L(s)  = 1  + (−0.777 − 0.777i)2-s + (1.60 − 0.664i)3-s + 0.209i·4-s + (0.171 + 0.413i)5-s + (−1.76 − 0.731i)6-s + (−0.499 + 1.20i)7-s + (−0.614 + 0.614i)8-s + (1.42 − 1.42i)9-s + (0.188 − 0.454i)10-s + (−1.18 − 0.489i)11-s + (0.139 + 0.336i)12-s + 0.0353i·13-s + (1.32 − 0.549i)14-s + (0.549 + 0.549i)15-s + 1.16·16-s + (−0.998 − 0.0468i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.294+0.955i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.294+0.955i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.294+0.955i0.294 + 0.955i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(36,⋅)\chi_{85} (36, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.294+0.955i)(2,\ 85,\ (\ :1/2),\ 0.294 + 0.955i)

Particular Values

L(1)L(1) ≈\approx 0.784527−0.578962i0.784527 - 0.578962i
L(12)L(\frac12) ≈\approx 0.784527−0.578962i0.784527 - 0.578962i
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)
bad5 1+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
17 1+(4.11+0.193i)T 1 + (4.11 + 0.193i)T
good2 1+(1.09+1.09i)T+2iT2 1 + (1.09 + 1.09i)T + 2iT^{2}
3 1+(−2.77+1.15i)T+(2.12−2.12i)T2 1 + (-2.77 + 1.15i)T + (2.12 - 2.12i)T^{2}
7 1+(1.32−3.19i)T+(−4.94−4.94i)T2 1 + (1.32 - 3.19i)T + (-4.94 - 4.94i)T^{2}
11 1+(3.92+1.62i)T+(7.77+7.77i)T2 1 + (3.92 + 1.62i)T + (7.77 + 7.77i)T^{2}
13 1−0.127iT−13T2 1 - 0.127iT - 13T^{2}
19 1+(−1.81−1.81i)T+19iT2 1 + (-1.81 - 1.81i)T + 19iT^{2}
23 1+(−3.00−1.24i)T+(16.2+16.2i)T2 1 + (-3.00 - 1.24i)T + (16.2 + 16.2i)T^{2}
29 1+(1.87+4.53i)T+(−20.5+20.5i)T2 1 + (1.87 + 4.53i)T + (-20.5 + 20.5i)T^{2}
31 1+(−4.95+2.05i)T+(21.9−21.9i)T2 1 + (-4.95 + 2.05i)T + (21.9 - 21.9i)T^{2}
37 1+(1.63−0.677i)T+(26.1−26.1i)T2 1 + (1.63 - 0.677i)T + (26.1 - 26.1i)T^{2}
41 1+(−3.85+9.29i)T+(−28.9−28.9i)T2 1 + (-3.85 + 9.29i)T + (-28.9 - 28.9i)T^{2}
43 1+(−1.79+1.79i)T−43iT2 1 + (-1.79 + 1.79i)T - 43iT^{2}
47 1−4.59iT−47T2 1 - 4.59iT - 47T^{2}
53 1+(−1.15−1.15i)T+53iT2 1 + (-1.15 - 1.15i)T + 53iT^{2}
59 1+(4.34−4.34i)T−59iT2 1 + (4.34 - 4.34i)T - 59iT^{2}
61 1+(1.54−3.73i)T+(−43.1−43.1i)T2 1 + (1.54 - 3.73i)T + (-43.1 - 43.1i)T^{2}
67 1+6.88T+67T2 1 + 6.88T + 67T^{2}
71 1+(6.66−2.76i)T+(50.2−50.2i)T2 1 + (6.66 - 2.76i)T + (50.2 - 50.2i)T^{2}
73 1+(5.59+13.5i)T+(−51.6+51.6i)T2 1 + (5.59 + 13.5i)T + (-51.6 + 51.6i)T^{2}
79 1+(4.75+1.97i)T+(55.8+55.8i)T2 1 + (4.75 + 1.97i)T + (55.8 + 55.8i)T^{2}
83 1+(−10.2−10.2i)T+83iT2 1 + (-10.2 - 10.2i)T + 83iT^{2}
89 1+0.600iT−89T2 1 + 0.600iT - 89T^{2}
97 1+(2.93+7.09i)T+(−68.5+68.5i)T2 1 + (2.93 + 7.09i)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

−13.89387786816538677169346709168, −13.03394027331866263843404685158, −11.89851705744823061792927714786, −10.50179204478189145769082746530, −9.330462709402958446569956007474, −8.745910606947320553786258201445, −7.65198205947563186366460164248, −5.94717550848239506590141142360, −2.97881761641922208990511247548, −2.26527285271161702860616254555, 3.06045011009649777364574448665, 4.53968901180004964880402790617, 6.98046322637125613878017282889, 7.84991687850485776524214418464, 8.817581727842394682961901378373, 9.684679158294848214187703316714, 10.51948075369233928363166974878, 12.94252736877124134952363595356, 13.46064690164771067306134289972, 14.73165090128531663686775856847

Graph of the ZZ-function along the critical line