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Properties

Label 2-55-5.2-c2-0-6
Degree 22
Conductor 5555
Sign 0.716+0.697i0.716 + 0.697i
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  + (0.188 + 0.188i)2-s + (0.812 − 0.812i)3-s − 3.92i·4-s + (−1.21 − 4.84i)5-s + 0.305·6-s + (5.94 + 5.94i)7-s + (1.49 − 1.49i)8-s + 7.67i·9-s + (0.683 − 1.14i)10-s − 3.31·11-s + (−3.19 − 3.19i)12-s + (3.02 − 3.02i)13-s + 2.23i·14-s + (−4.92 − 2.95i)15-s − 15.1·16-s + (0.878 + 0.878i)17-s + ⋯
L(s)  = 1  + (0.0940 + 0.0940i)2-s + (0.270 − 0.270i)3-s − 0.982i·4-s + (−0.243 − 0.969i)5-s + 0.0509·6-s + (0.849 + 0.849i)7-s + (0.186 − 0.186i)8-s + 0.853i·9-s + (0.0683 − 0.114i)10-s − 0.301·11-s + (−0.266 − 0.266i)12-s + (0.232 − 0.232i)13-s + 0.159i·14-s + (−0.328 − 0.196i)15-s − 0.947·16-s + (0.0516 + 0.0516i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.716+0.697i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1) L(s)=((0.716+0.697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.716+0.697i0.716 + 0.697i
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), 0.716+0.697i)(2,\ 55,\ (\ :1),\ 0.716 + 0.697i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.21584−0.493797i1.21584 - 0.493797i
L(12)L(\frac12) ≈\approx 1.21584−0.493797i1.21584 - 0.493797i
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+(1.21+4.84i)T 1 + (1.21 + 4.84i)T
11 1+3.31T 1 + 3.31T
good2 1+(−0.188−0.188i)T+4iT2 1 + (-0.188 - 0.188i)T + 4iT^{2}
3 1+(−0.812+0.812i)T−9iT2 1 + (-0.812 + 0.812i)T - 9iT^{2}
7 1+(−5.94−5.94i)T+49iT2 1 + (-5.94 - 5.94i)T + 49iT^{2}
13 1+(−3.02+3.02i)T−169iT2 1 + (-3.02 + 3.02i)T - 169iT^{2}
17 1+(−0.878−0.878i)T+289iT2 1 + (-0.878 - 0.878i)T + 289iT^{2}
19 1−29.2iT−361T2 1 - 29.2iT - 361T^{2}
23 1+(−11.8+11.8i)T−529iT2 1 + (-11.8 + 11.8i)T - 529iT^{2}
29 1+15.7iT−841T2 1 + 15.7iT - 841T^{2}
31 1+14.3T+961T2 1 + 14.3T + 961T^{2}
37 1+(−36.1−36.1i)T+1.36e3iT2 1 + (-36.1 - 36.1i)T + 1.36e3iT^{2}
41 1+57.6T+1.68e3T2 1 + 57.6T + 1.68e3T^{2}
43 1+(−58.0+58.0i)T−1.84e3iT2 1 + (-58.0 + 58.0i)T - 1.84e3iT^{2}
47 1+(29.8+29.8i)T+2.20e3iT2 1 + (29.8 + 29.8i)T + 2.20e3iT^{2}
53 1+(59.0−59.0i)T−2.80e3iT2 1 + (59.0 - 59.0i)T - 2.80e3iT^{2}
59 1+61.7iT−3.48e3T2 1 + 61.7iT - 3.48e3T^{2}
61 1−1.59T+3.72e3T2 1 - 1.59T + 3.72e3T^{2}
67 1+(32.7+32.7i)T+4.48e3iT2 1 + (32.7 + 32.7i)T + 4.48e3iT^{2}
71 1+12.4T+5.04e3T2 1 + 12.4T + 5.04e3T^{2}
73 1+(74.3−74.3i)T−5.32e3iT2 1 + (74.3 - 74.3i)T - 5.32e3iT^{2}
79 1+21.8iT−6.24e3T2 1 + 21.8iT - 6.24e3T^{2}
83 1+(−45.9+45.9i)T−6.88e3iT2 1 + (-45.9 + 45.9i)T - 6.88e3iT^{2}
89 1+89.4iT−7.92e3T2 1 + 89.4iT - 7.92e3T^{2}
97 1+(62.6+62.6i)T+9.40e3iT2 1 + (62.6 + 62.6i)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

−14.91236514104645012886838623867, −13.89562752488202578059572719019, −12.77653972500990229084946411204, −11.53225653797741929448524672565, −10.26162891416412529553872673071, −8.776852003322868304135469040473, −7.86275404361550842304064051547, −5.75530122640673775529865152377, −4.79793864925451289393078551130, −1.76683815116661996695888395739, 3.08384559203806056997747944258, 4.36525013173734177040317102577, 6.85931778670459618456055619339, 7.80201233235856823630566270415, 9.214108985083621446152743963159, 10.88201903857509299940240571856, 11.55016993275592071745706803912, 13.04694667949253652209071881631, 14.15119820644746608436382645840, 15.08937587346757787245243660242

Graph of the ZZ-function along the critical line