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Properties

Label 2-55-5.2-c2-0-8
Degree 22
Conductor 5555
Sign −0.967+0.253i-0.967 + 0.253i
Analytic cond. 1.498641.49864
Root an. cond. 1.224191.22419
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.864 − 0.864i)2-s + (−2.45 + 2.45i)3-s − 2.50i·4-s + (−4.77 − 1.46i)5-s + 4.24·6-s + (−6.72 − 6.72i)7-s + (−5.62 + 5.62i)8-s − 3.03i·9-s + (2.86 + 5.40i)10-s + 3.31·11-s + (6.14 + 6.14i)12-s + (0.519 − 0.519i)13-s + 11.6i·14-s + (15.3 − 8.12i)15-s − 0.288·16-s + (5.04 + 5.04i)17-s + ⋯
L(s)  = 1  + (−0.432 − 0.432i)2-s + (−0.817 + 0.817i)3-s − 0.626i·4-s + (−0.955 − 0.293i)5-s + 0.707·6-s + (−0.960 − 0.960i)7-s + (−0.703 + 0.703i)8-s − 0.337i·9-s + (0.286 + 0.540i)10-s + 0.301·11-s + (0.511 + 0.511i)12-s + (0.0399 − 0.0399i)13-s + 0.831i·14-s + (1.02 − 0.541i)15-s − 0.0180·16-s + (0.296 + 0.296i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((−0.967+0.253i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1) L(s)=((−0.967+0.253i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: −0.967+0.253i-0.967 + 0.253i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(12,⋅)\chi_{55} (12, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), −0.967+0.253i)(2,\ 55,\ (\ :1),\ -0.967 + 0.253i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0285833−0.222238i0.0285833 - 0.222238i
L(12)L(\frac12) ≈\approx 0.0285833−0.222238i0.0285833 - 0.222238i
L(2)L(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+(4.77+1.46i)T 1 + (4.77 + 1.46i)T
11 1−3.31T 1 - 3.31T
good2 1+(0.864+0.864i)T+4iT2 1 + (0.864 + 0.864i)T + 4iT^{2}
3 1+(2.45−2.45i)T−9iT2 1 + (2.45 - 2.45i)T - 9iT^{2}
7 1+(6.72+6.72i)T+49iT2 1 + (6.72 + 6.72i)T + 49iT^{2}
13 1+(−0.519+0.519i)T−169iT2 1 + (-0.519 + 0.519i)T - 169iT^{2}
17 1+(−5.04−5.04i)T+289iT2 1 + (-5.04 - 5.04i)T + 289iT^{2}
19 1+25.5iT−361T2 1 + 25.5iT - 361T^{2}
23 1+(5.12−5.12i)T−529iT2 1 + (5.12 - 5.12i)T - 529iT^{2}
29 1−0.0328iT−841T2 1 - 0.0328iT - 841T^{2}
31 1+51.6T+961T2 1 + 51.6T + 961T^{2}
37 1+(17.2+17.2i)T+1.36e3iT2 1 + (17.2 + 17.2i)T + 1.36e3iT^{2}
41 1+26.0T+1.68e3T2 1 + 26.0T + 1.68e3T^{2}
43 1+(−49.5+49.5i)T−1.84e3iT2 1 + (-49.5 + 49.5i)T - 1.84e3iT^{2}
47 1+(60.9+60.9i)T+2.20e3iT2 1 + (60.9 + 60.9i)T + 2.20e3iT^{2}
53 1+(−19.8+19.8i)T−2.80e3iT2 1 + (-19.8 + 19.8i)T - 2.80e3iT^{2}
59 1−108.iT−3.48e3T2 1 - 108. iT - 3.48e3T^{2}
61 1−103.T+3.72e3T2 1 - 103.T + 3.72e3T^{2}
67 1+(−13.8−13.8i)T+4.48e3iT2 1 + (-13.8 - 13.8i)T + 4.48e3iT^{2}
71 1+65.5T+5.04e3T2 1 + 65.5T + 5.04e3T^{2}
73 1+(39.0−39.0i)T−5.32e3iT2 1 + (39.0 - 39.0i)T - 5.32e3iT^{2}
79 1+29.1iT−6.24e3T2 1 + 29.1iT - 6.24e3T^{2}
83 1+(−55.8+55.8i)T−6.88e3iT2 1 + (-55.8 + 55.8i)T - 6.88e3iT^{2}
89 1+6.65iT−7.92e3T2 1 + 6.65iT - 7.92e3T^{2}
97 1+(−34.5−34.5i)T+9.40e3iT2 1 + (-34.5 - 34.5i)T + 9.40e3iT^{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

−14.86290137739138155552243778421, −13.29515505810438623522379704878, −11.79982164427181389041029411870, −10.89715325497930999782491066579, −10.13130569186413203864492271296, −8.966258078616924229922912789623, −7.02955156488428508665474948536, −5.36203099952563642872309806974, −3.88486998208537043868867333864, −0.26922146434579601786721871911, 3.44658421505604742158215086878, 6.03591269644729694775129567954, 6.98666149012794671939772920991, 8.120471924500554429036506050937, 9.466985049766085731114693360009, 11.39702728454446698401163882642, 12.33461723522029170180149439981, 12.71074445787843264415932834500, 14.73901181932060958000327951350, 16.00289099678856871470213797269

Graph of the ZZ-function along the critical line