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Properties

Label 2-55-55.18-c1-0-0
Degree 22
Conductor 5555
Sign −0.104−0.994i-0.104 - 0.994i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (0.193 + 1.22i)2-s + (−2.26 + 1.15i)3-s + (0.443 − 0.144i)4-s + (−0.536 + 2.17i)5-s + (−1.85 − 2.54i)6-s + (1.57 − 3.09i)7-s + (1.38 + 2.72i)8-s + (2.04 − 2.80i)9-s + (−2.75 − 0.235i)10-s + (0.937 − 3.18i)11-s + (−0.839 + 0.839i)12-s + (0.730 − 0.115i)13-s + (4.08 + 1.32i)14-s + (−1.29 − 5.54i)15-s + (−2.30 + 1.67i)16-s + (−0.609 − 0.0965i)17-s + ⋯
L(s)  = 1  + (0.136 + 0.864i)2-s + (−1.30 + 0.666i)3-s + (0.221 − 0.0720i)4-s + (−0.239 + 0.970i)5-s + (−0.755 − 1.04i)6-s + (0.595 − 1.16i)7-s + (0.490 + 0.962i)8-s + (0.680 − 0.936i)9-s + (−0.872 − 0.0744i)10-s + (0.282 − 0.959i)11-s + (−0.242 + 0.242i)12-s + (0.202 − 0.0320i)13-s + (1.09 + 0.354i)14-s + (−0.333 − 1.43i)15-s + (−0.576 + 0.418i)16-s + (−0.147 − 0.0234i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((−0.104−0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((−0.104−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: −0.104−0.994i-0.104 - 0.994i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(18,⋅)\chi_{55} (18, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), −0.104−0.994i)(2,\ 55,\ (\ :1/2),\ -0.104 - 0.994i)

Particular Values

L(1)L(1) ≈\approx 0.490778+0.545097i0.490778 + 0.545097i
L(12)L(\frac12) ≈\approx 0.490778+0.545097i0.490778 + 0.545097i
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+(0.536−2.17i)T 1 + (0.536 - 2.17i)T
11 1+(−0.937+3.18i)T 1 + (-0.937 + 3.18i)T
good2 1+(−0.193−1.22i)T+(−1.90+0.618i)T2 1 + (-0.193 - 1.22i)T + (-1.90 + 0.618i)T^{2}
3 1+(2.26−1.15i)T+(1.76−2.42i)T2 1 + (2.26 - 1.15i)T + (1.76 - 2.42i)T^{2}
7 1+(−1.57+3.09i)T+(−4.11−5.66i)T2 1 + (-1.57 + 3.09i)T + (-4.11 - 5.66i)T^{2}
13 1+(−0.730+0.115i)T+(12.3−4.01i)T2 1 + (-0.730 + 0.115i)T + (12.3 - 4.01i)T^{2}
17 1+(0.609+0.0965i)T+(16.1+5.25i)T2 1 + (0.609 + 0.0965i)T + (16.1 + 5.25i)T^{2}
19 1+(0.971−2.99i)T+(−15.3−11.1i)T2 1 + (0.971 - 2.99i)T + (-15.3 - 11.1i)T^{2}
23 1+(4.30+4.30i)T+23iT2 1 + (4.30 + 4.30i)T + 23iT^{2}
29 1+(−0.896−2.75i)T+(−23.4+17.0i)T2 1 + (-0.896 - 2.75i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.45+1.78i)T+(9.57+29.4i)T2 1 + (2.45 + 1.78i)T + (9.57 + 29.4i)T^{2}
37 1+(−2.04−1.04i)T+(21.7+29.9i)T2 1 + (-2.04 - 1.04i)T + (21.7 + 29.9i)T^{2}
41 1+(−0.970−0.315i)T+(33.1+24.0i)T2 1 + (-0.970 - 0.315i)T + (33.1 + 24.0i)T^{2}
43 1+(−4.07+4.07i)T−43iT2 1 + (-4.07 + 4.07i)T - 43iT^{2}
47 1+(0.492+0.967i)T+(−27.6+38.0i)T2 1 + (0.492 + 0.967i)T + (-27.6 + 38.0i)T^{2}
53 1+(0.671+4.24i)T+(−50.4+16.3i)T2 1 + (0.671 + 4.24i)T + (-50.4 + 16.3i)T^{2}
59 1+(7.03−2.28i)T+(47.7−34.6i)T2 1 + (7.03 - 2.28i)T + (47.7 - 34.6i)T^{2}
61 1+(2.20+3.03i)T+(−18.8+58.0i)T2 1 + (2.20 + 3.03i)T + (-18.8 + 58.0i)T^{2}
67 1+(9.39−9.39i)T−67iT2 1 + (9.39 - 9.39i)T - 67iT^{2}
71 1+(2.92−2.12i)T+(21.9−67.5i)T2 1 + (2.92 - 2.12i)T + (21.9 - 67.5i)T^{2}
73 1+(0.479+0.244i)T+(42.9+59.0i)T2 1 + (0.479 + 0.244i)T + (42.9 + 59.0i)T^{2}
79 1+(−9.87−7.17i)T+(24.4+75.1i)T2 1 + (-9.87 - 7.17i)T + (24.4 + 75.1i)T^{2}
83 1+(2.50−15.8i)T+(−78.9−25.6i)T2 1 + (2.50 - 15.8i)T + (-78.9 - 25.6i)T^{2}
89 1+14.6iT−89T2 1 + 14.6iT - 89T^{2}
97 1+(−14.1+2.24i)T+(92.2−29.9i)T2 1 + (-14.1 + 2.24i)T + (92.2 - 29.9i)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

−15.89259832763877706051763158298, −14.65413308887404888315656224707, −13.92220949713260417919998689891, −11.74948806620884617055424470831, −10.86936945921835702540106847633, −10.44607456939842040682535768156, −7.972168957312376680585780671014, −6.70492514948441848543379614814, −5.78679632060962614644794923866, −4.19362398092889181747246390347, 1.75439518332264450609609235534, 4.65390929457750357430755361487, 6.03423753603751555224832820628, 7.58517741399089727199820945541, 9.307876682657034053209906588818, 10.98083249162101526466197397520, 11.95663211049707511257133762512, 12.20494485567282101624870350723, 13.22053364856700552030768652736, 15.30024077763471424241189454443

Graph of the ZZ-function along the critical line