↑

Properties

Label 2-272-16.13-c1-0-30
Degree 22
Conductor 272272
Sign −0.865−0.500i-0.865 - 0.500i
Analytic cond. 2.171932.17193
Root an. cond. 1.473741.47374
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.321 − 1.37i)2-s + (−2.20 − 2.20i)3-s + (−1.79 − 0.885i)4-s + (2.24 − 2.24i)5-s + (−3.74 + 2.33i)6-s − 3.15i·7-s + (−1.79 + 2.18i)8-s + 6.74i·9-s + (−2.36 − 3.80i)10-s + (2.18 − 2.18i)11-s + (2.00 + 5.91i)12-s + (1.93 + 1.93i)13-s + (−4.34 − 1.01i)14-s − 9.89·15-s + (2.43 + 3.17i)16-s − 17-s + ⋯
L(s)  = 1  + (0.227 − 0.973i)2-s + (−1.27 − 1.27i)3-s + (−0.896 − 0.442i)4-s + (1.00 − 1.00i)5-s + (−1.53 + 0.951i)6-s − 1.19i·7-s + (−0.634 + 0.772i)8-s + 2.24i·9-s + (−0.748 − 1.20i)10-s + (0.659 − 0.659i)11-s + (0.578 + 1.70i)12-s + (0.536 + 0.536i)13-s + (−1.16 − 0.270i)14-s − 2.55·15-s + (0.608 + 0.793i)16-s − 0.242·17-s + ⋯

Functional equation

Λ(s)=(272s/2 ΓC(s) L(s)=((−0.865−0.500i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(272s/2 ΓC(s+1/2) L(s)=((−0.865−0.500i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 272272    =    24⋅172^{4} \cdot 17
Sign: −0.865−0.500i-0.865 - 0.500i
Analytic conductor: 2.171932.17193
Root analytic conductor: 1.473741.47374
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ272(205,⋅)\chi_{272} (205, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 272, ( :1/2), −0.865−0.500i)(2,\ 272,\ (\ :1/2),\ -0.865 - 0.500i)

Particular Values

L(1)L(1) ≈\approx 0.259582+0.967785i0.259582 + 0.967785i
L(12)L(\frac12) ≈\approx 0.259582+0.967785i0.259582 + 0.967785i
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)
bad2 1+(−0.321+1.37i)T 1 + (-0.321 + 1.37i)T
17 1+T 1 + T
good3 1+(2.20+2.20i)T+3iT2 1 + (2.20 + 2.20i)T + 3iT^{2}
5 1+(−2.24+2.24i)T−5iT2 1 + (-2.24 + 2.24i)T - 5iT^{2}
7 1+3.15iT−7T2 1 + 3.15iT - 7T^{2}
11 1+(−2.18+2.18i)T−11iT2 1 + (-2.18 + 2.18i)T - 11iT^{2}
13 1+(−1.93−1.93i)T+13iT2 1 + (-1.93 - 1.93i)T + 13iT^{2}
19 1+(−5.15−5.15i)T+19iT2 1 + (-5.15 - 5.15i)T + 19iT^{2}
23 1−0.195iT−23T2 1 - 0.195iT - 23T^{2}
29 1+(0.528+0.528i)T+29iT2 1 + (0.528 + 0.528i)T + 29iT^{2}
31 1+6.11T+31T2 1 + 6.11T + 31T^{2}
37 1+(−1.40+1.40i)T−37iT2 1 + (-1.40 + 1.40i)T - 37iT^{2}
41 1+4.43iT−41T2 1 + 4.43iT - 41T^{2}
43 1+(0.190−0.190i)T−43iT2 1 + (0.190 - 0.190i)T - 43iT^{2}
47 1−3.42T+47T2 1 - 3.42T + 47T^{2}
53 1+(−8.10+8.10i)T−53iT2 1 + (-8.10 + 8.10i)T - 53iT^{2}
59 1+(8.44−8.44i)T−59iT2 1 + (8.44 - 8.44i)T - 59iT^{2}
61 1+(−0.714−0.714i)T+61iT2 1 + (-0.714 - 0.714i)T + 61iT^{2}
67 1+(6.97+6.97i)T+67iT2 1 + (6.97 + 6.97i)T + 67iT^{2}
71 1−12.1iT−71T2 1 - 12.1iT - 71T^{2}
73 1−9.75iT−73T2 1 - 9.75iT - 73T^{2}
79 1+12.9T+79T2 1 + 12.9T + 79T^{2}
83 1+(−6.54−6.54i)T+83iT2 1 + (-6.54 - 6.54i)T + 83iT^{2}
89 1−5.65iT−89T2 1 - 5.65iT - 89T^{2}
97 1+18.8T+97T2 1 + 18.8T + 97T^{2}
show more
show less
   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

−11.50127978091883024946120104169, −10.73694513217635141913551001171, −9.734704704454331918774967624296, −8.601480743413030864785451491856, −7.23345533862900802013767982298, −5.97392123218435831219824644781, −5.41196635385728221687447931613, −4.03196180960064179914404176818, −1.65833379239036168267058544058, −0.955266944501218711267585836350, 3.17383907648099418112129262623, 4.66458404524654967041682103905, 5.63177123324909353926813177622, 6.11952905704649338545572882332, 7.09568395564861925004533403509, 9.073471386877672523887486611005, 9.483859761475350230754371490861, 10.46758861853251442718960487261, 11.45303441457067202391356919966, 12.32575819561799111851511844285

Graph of the ZZ-function along the critical line