Properties

Label 2-55-55.28-c1-0-2
Degree 22
Conductor 5555
Sign 0.995+0.0917i0.995 + 0.0917i
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  + (−0.474 + 0.930i)2-s + (−0.440 − 2.78i)3-s + (0.533 + 0.734i)4-s + (2.23 − 0.0540i)5-s + (2.79 + 0.909i)6-s + (−0.543 − 0.0860i)7-s + (−3.00 + 0.475i)8-s + (−4.68 + 1.52i)9-s + (−1.01 + 2.10i)10-s + (−3.29 + 0.335i)11-s + (1.80 − 1.80i)12-s + (2.89 + 1.47i)13-s + (0.337 − 0.464i)14-s + (−1.13 − 6.19i)15-s + (0.419 − 1.29i)16-s + (−5.04 + 2.57i)17-s + ⋯
L(s)  = 1  + (−0.335 + 0.658i)2-s + (−0.254 − 1.60i)3-s + (0.266 + 0.367i)4-s + (0.999 − 0.0241i)5-s + (1.14 + 0.371i)6-s + (−0.205 − 0.0325i)7-s + (−1.06 + 0.168i)8-s + (−1.56 + 0.507i)9-s + (−0.319 + 0.666i)10-s + (−0.994 + 0.101i)11-s + (0.522 − 0.522i)12-s + (0.801 + 0.408i)13-s + (0.0902 − 0.124i)14-s + (−0.293 − 1.59i)15-s + (0.104 − 0.322i)16-s + (−1.22 + 0.623i)17-s + ⋯

Functional equation

Λ(s)=(55s/2ΓC(s)L(s)=((0.995+0.0917i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2ΓC(s+1/2)L(s)=((0.995+0.0917i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5115 \cdot 11
Sign: 0.995+0.0917i0.995 + 0.0917i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(28,)\chi_{55} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.995+0.0917i)(2,\ 55,\ (\ :1/2),\ 0.995 + 0.0917i)

Particular Values

L(1)L(1) \approx 0.7693820.0353856i0.769382 - 0.0353856i
L(12)L(\frac12) \approx 0.7693820.0353856i0.769382 - 0.0353856i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(2.23+0.0540i)T 1 + (-2.23 + 0.0540i)T
11 1+(3.290.335i)T 1 + (3.29 - 0.335i)T
good2 1+(0.4740.930i)T+(1.171.61i)T2 1 + (0.474 - 0.930i)T + (-1.17 - 1.61i)T^{2}
3 1+(0.440+2.78i)T+(2.85+0.927i)T2 1 + (0.440 + 2.78i)T + (-2.85 + 0.927i)T^{2}
7 1+(0.543+0.0860i)T+(6.65+2.16i)T2 1 + (0.543 + 0.0860i)T + (6.65 + 2.16i)T^{2}
13 1+(2.891.47i)T+(7.64+10.5i)T2 1 + (-2.89 - 1.47i)T + (7.64 + 10.5i)T^{2}
17 1+(5.042.57i)T+(9.9913.7i)T2 1 + (5.04 - 2.57i)T + (9.99 - 13.7i)T^{2}
19 1+(1.250.914i)T+(5.87+18.0i)T2 1 + (-1.25 - 0.914i)T + (5.87 + 18.0i)T^{2}
23 1+(0.803+0.803i)T+23iT2 1 + (0.803 + 0.803i)T + 23iT^{2}
29 1+(3.44+2.50i)T+(8.9627.5i)T2 1 + (-3.44 + 2.50i)T + (8.96 - 27.5i)T^{2}
31 1+(0.509+1.56i)T+(25.0+18.2i)T2 1 + (0.509 + 1.56i)T + (-25.0 + 18.2i)T^{2}
37 1+(0.1490.945i)T+(35.111.4i)T2 1 + (0.149 - 0.945i)T + (-35.1 - 11.4i)T^{2}
41 1+(5.25+7.23i)T+(12.638.9i)T2 1 + (-5.25 + 7.23i)T + (-12.6 - 38.9i)T^{2}
43 1+(2.55+2.55i)T43iT2 1 + (-2.55 + 2.55i)T - 43iT^{2}
47 1+(4.02+0.636i)T+(44.614.5i)T2 1 + (-4.02 + 0.636i)T + (44.6 - 14.5i)T^{2}
53 1+(3.196.27i)T+(31.142.8i)T2 1 + (3.19 - 6.27i)T + (-31.1 - 42.8i)T^{2}
59 1+(3.975.47i)T+(18.2+56.1i)T2 1 + (-3.97 - 5.47i)T + (-18.2 + 56.1i)T^{2}
61 1+(8.75+2.84i)T+(49.3+35.8i)T2 1 + (8.75 + 2.84i)T + (49.3 + 35.8i)T^{2}
67 1+(2.622.62i)T67iT2 1 + (2.62 - 2.62i)T - 67iT^{2}
71 1+(2.11+6.51i)T+(57.441.7i)T2 1 + (-2.11 + 6.51i)T + (-57.4 - 41.7i)T^{2}
73 1+(1.57+9.96i)T+(69.422.5i)T2 1 + (-1.57 + 9.96i)T + (-69.4 - 22.5i)T^{2}
79 1+(1.283.96i)T+(63.9+46.4i)T2 1 + (-1.28 - 3.96i)T + (-63.9 + 46.4i)T^{2}
83 1+(5.1410.0i)T+(48.7+67.1i)T2 1 + (-5.14 - 10.0i)T + (-48.7 + 67.1i)T^{2}
89 13.64iT89T2 1 - 3.64iT - 89T^{2}
97 1+(14.87.56i)T+(57.0+78.4i)T2 1 + (-14.8 - 7.56i)T + (57.0 + 78.4i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−15.51754999881598381413867315642, −13.87455343802352467968095923407, −13.11349212285347358121412436226, −12.23867769931468976673391616349, −10.83980658820952966517355792890, −8.923872544738054384216099954414, −7.80962763772320751283150377086, −6.64131331392991508178295589584, −5.91964563116307206628612997750, −2.30201715349521540754721535038, 2.90621412093884840572391895095, 5.03398526570061603130591842774, 6.18452953024848744760122570952, 8.926292763870742986135841980245, 9.776804341246773447516730564630, 10.61413970828910872431206934616, 11.25440439795185372817309568595, 13.06871289874405598854395185784, 14.43294072849954603034576975303, 15.68791991476255340378338810510

Graph of the ZZ-function along the critical line