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Properties

Label 2-85-17.13-c1-0-3
Degree 22
Conductor 8585
Sign 0.471+0.881i0.471 + 0.881i
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  − 2.07i·2-s + (1.78 + 1.78i)3-s − 2.28·4-s + (−0.707 − 0.707i)5-s + (3.69 − 3.69i)6-s + (−0.260 + 0.260i)7-s + 0.595i·8-s + 3.36i·9-s + (−1.46 + 1.46i)10-s + (−1.76 + 1.76i)11-s + (−4.08 − 4.08i)12-s − 4.68·13-s + (0.540 + 0.540i)14-s − 2.52i·15-s − 3.34·16-s + (3.84 − 1.48i)17-s + ⋯
L(s)  = 1  − 1.46i·2-s + (1.03 + 1.03i)3-s − 1.14·4-s + (−0.316 − 0.316i)5-s + (1.50 − 1.50i)6-s + (−0.0986 + 0.0986i)7-s + 0.210i·8-s + 1.12i·9-s + (−0.463 + 0.463i)10-s + (−0.532 + 0.532i)11-s + (−1.17 − 1.17i)12-s − 1.30·13-s + (0.144 + 0.144i)14-s − 0.651i·15-s − 0.835·16-s + (0.932 − 0.360i)17-s + ⋯

Functional equation

Λ(s)=(85s/2 ΓC(s) L(s)=((0.471+0.881i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(85s/2 ΓC(s+1/2) L(s)=((0.471+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8585    =    5⋅175 \cdot 17
Sign: 0.471+0.881i0.471 + 0.881i
Analytic conductor: 0.6787280.678728
Root analytic conductor: 0.8238490.823849
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ85(81,⋅)\chi_{85} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 85, ( :1/2), 0.471+0.881i)(2,\ 85,\ (\ :1/2),\ 0.471 + 0.881i)

Particular Values

L(1)L(1) ≈\approx 0.987545−0.591503i0.987545 - 0.591503i
L(12)L(\frac12) ≈\approx 0.987545−0.591503i0.987545 - 0.591503i
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
17 1+(−3.84+1.48i)T 1 + (-3.84 + 1.48i)T
good2 1+2.07iT−2T2 1 + 2.07iT - 2T^{2}
3 1+(−1.78−1.78i)T+3iT2 1 + (-1.78 - 1.78i)T + 3iT^{2}
7 1+(0.260−0.260i)T−7iT2 1 + (0.260 - 0.260i)T - 7iT^{2}
11 1+(1.76−1.76i)T−11iT2 1 + (1.76 - 1.76i)T - 11iT^{2}
13 1+4.68T+13T2 1 + 4.68T + 13T^{2}
19 1−7.16iT−19T2 1 - 7.16iT - 19T^{2}
23 1+(−4.73+4.73i)T−23iT2 1 + (-4.73 + 4.73i)T - 23iT^{2}
29 1+(4.79+4.79i)T+29iT2 1 + (4.79 + 4.79i)T + 29iT^{2}
31 1+(−3.40−3.40i)T+31iT2 1 + (-3.40 - 3.40i)T + 31iT^{2}
37 1+(1.37+1.37i)T+37iT2 1 + (1.37 + 1.37i)T + 37iT^{2}
41 1+(−1.66+1.66i)T−41iT2 1 + (-1.66 + 1.66i)T - 41iT^{2}
43 1+11.7iT−43T2 1 + 11.7iT - 43T^{2}
47 1+1.65T+47T2 1 + 1.65T + 47T^{2}
53 1−6.81iT−53T2 1 - 6.81iT - 53T^{2}
59 1+0.484iT−59T2 1 + 0.484iT - 59T^{2}
61 1+(−4.86+4.86i)T−61iT2 1 + (-4.86 + 4.86i)T - 61iT^{2}
67 1+1.87T+67T2 1 + 1.87T + 67T^{2}
71 1+(−1.21−1.21i)T+71iT2 1 + (-1.21 - 1.21i)T + 71iT^{2}
73 1+(−0.202−0.202i)T+73iT2 1 + (-0.202 - 0.202i)T + 73iT^{2}
79 1+(−3.80+3.80i)T−79iT2 1 + (-3.80 + 3.80i)T - 79iT^{2}
83 1+9.94iT−83T2 1 + 9.94iT - 83T^{2}
89 1+4.30T+89T2 1 + 4.30T + 89T^{2}
97 1+(−9.01−9.01i)T+97iT2 1 + (-9.01 - 9.01i)T + 97iT^{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

−14.06128522225399863656765103072, −12.64170636757837624318781762062, −12.03557549874680266361706248384, −10.45800332988412824064329003933, −9.916671992158567678504254953536, −9.000528741146007901423530254936, −7.66026382121540309232587253662, −4.89236162483093334867968802127, −3.72085121446463978584232695108, −2.50606871569465001846473175196, 2.86124431411402519785983983850, 5.19449471782889344578460285821, 6.82466121839099380516016418398, 7.47767565254877150870191224753, 8.266036093035617607070625286636, 9.444125492648529809661950699553, 11.33766067025112354811655048670, 12.90048161249393624089736919365, 13.61531947705170210897344216945, 14.68612910734167593333995276869

Graph of the ZZ-function along the critical line