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Properties

Label 2-272-16.5-c1-0-30
Degree 22
Conductor 272272
Sign −0.981+0.189i-0.981 + 0.189i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.205 − 1.39i)2-s + (1.44 − 1.44i)3-s + (−1.91 + 0.574i)4-s + (−1.55 − 1.55i)5-s + (−2.31 − 1.72i)6-s − 2.87i·7-s + (1.19 + 2.56i)8-s − 1.17i·9-s + (−1.85 + 2.49i)10-s + (0.709 + 0.709i)11-s + (−1.93 + 3.59i)12-s + (−1.15 + 1.15i)13-s + (−4.02 + 0.590i)14-s − 4.49·15-s + (3.33 − 2.20i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.145 − 0.989i)2-s + (0.834 − 0.834i)3-s + (−0.957 + 0.287i)4-s + (−0.694 − 0.694i)5-s + (−0.946 − 0.704i)6-s − 1.08i·7-s + (0.423 + 0.905i)8-s − 0.392i·9-s + (−0.586 + 0.788i)10-s + (0.213 + 0.213i)11-s + (−0.559 + 1.03i)12-s + (−0.320 + 0.320i)13-s + (−1.07 + 0.157i)14-s − 1.15·15-s + (0.834 − 0.550i)16-s − 0.242·17-s + ⋯

Functional equation

Λ(s)=(272s/2 ΓC(s) L(s)=((−0.981+0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(272s/2 ΓC(s+1/2) L(s)=((−0.981+0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 272272    =    24⋅172^{4} \cdot 17
Sign: −0.981+0.189i-0.981 + 0.189i
Analytic conductor: 2.171932.17193
Root analytic conductor: 1.473741.47374
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ272(69,⋅)\chi_{272} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 272, ( :1/2), −0.981+0.189i)(2,\ 272,\ (\ :1/2),\ -0.981 + 0.189i)

Particular Values

L(1)L(1) ≈\approx 0.111385−1.16726i0.111385 - 1.16726i
L(12)L(\frac12) ≈\approx 0.111385−1.16726i0.111385 - 1.16726i
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+(0.205+1.39i)T 1 + (0.205 + 1.39i)T
17 1+T 1 + T
good3 1+(−1.44+1.44i)T−3iT2 1 + (-1.44 + 1.44i)T - 3iT^{2}
5 1+(1.55+1.55i)T+5iT2 1 + (1.55 + 1.55i)T + 5iT^{2}
7 1+2.87iT−7T2 1 + 2.87iT - 7T^{2}
11 1+(−0.709−0.709i)T+11iT2 1 + (-0.709 - 0.709i)T + 11iT^{2}
13 1+(1.15−1.15i)T−13iT2 1 + (1.15 - 1.15i)T - 13iT^{2}
19 1+(−0.443+0.443i)T−19iT2 1 + (-0.443 + 0.443i)T - 19iT^{2}
23 1+3.88iT−23T2 1 + 3.88iT - 23T^{2}
29 1+(−1.60+1.60i)T−29iT2 1 + (-1.60 + 1.60i)T - 29iT^{2}
31 1−3.14T+31T2 1 - 3.14T + 31T^{2}
37 1+(−1.86−1.86i)T+37iT2 1 + (-1.86 - 1.86i)T + 37iT^{2}
41 1+10.3iT−41T2 1 + 10.3iT - 41T^{2}
43 1+(−8.83−8.83i)T+43iT2 1 + (-8.83 - 8.83i)T + 43iT^{2}
47 1+8.60T+47T2 1 + 8.60T + 47T^{2}
53 1+(3.15+3.15i)T+53iT2 1 + (3.15 + 3.15i)T + 53iT^{2}
59 1+(−7.60−7.60i)T+59iT2 1 + (-7.60 - 7.60i)T + 59iT^{2}
61 1+(−7.69+7.69i)T−61iT2 1 + (-7.69 + 7.69i)T - 61iT^{2}
67 1+(−2.54+2.54i)T−67iT2 1 + (-2.54 + 2.54i)T - 67iT^{2}
71 1+2.42iT−71T2 1 + 2.42iT - 71T^{2}
73 1−11.8iT−73T2 1 - 11.8iT - 73T^{2}
79 1−13.0T+79T2 1 - 13.0T + 79T^{2}
83 1+(12.1−12.1i)T−83iT2 1 + (12.1 - 12.1i)T - 83iT^{2}
89 1+13.5iT−89T2 1 + 13.5iT - 89T^{2}
97 1+2.79T+97T2 1 + 2.79T + 97T^{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

−11.60466452572310038205167742214, −10.58715834313986812500911113239, −9.532582656088283015834160792201, −8.475091127865437472120332177778, −7.899388018756021339378650306624, −6.90281192267614103969503984731, −4.74500633100781755173862474039, −3.90715692416065188093107104309, −2.43780613374890356806691810726, −0.935022316026765515711472153870, 2.97640871997520696575982379302, 4.01758557240042403262845103661, 5.29222841786532611397142437418, 6.48838371465730216729438936726, 7.66346253751685234559883981757, 8.513866446784124305036070667381, 9.286281417927296554766090472178, 10.07138583303634431764358421702, 11.30603260466159877270443779722, 12.41127128489345549242094509448

Graph of the ZZ-function along the critical line