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Properties

Label 2-272-16.13-c1-0-25
Degree 22
Conductor 272272
Sign 0.235+0.971i0.235 + 0.971i
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  + (0.591 − 1.28i)2-s + (0.915 + 0.915i)3-s + (−1.30 − 1.51i)4-s + (2.93 − 2.93i)5-s + (1.71 − 0.634i)6-s + 1.91i·7-s + (−2.72 + 0.771i)8-s − 1.32i·9-s + (−2.03 − 5.51i)10-s + (−2.96 + 2.96i)11-s + (0.201 − 2.58i)12-s + (1.16 + 1.16i)13-s + (2.45 + 1.13i)14-s + 5.37·15-s + (−0.619 + 3.95i)16-s + 17-s + ⋯
L(s)  = 1  + (0.418 − 0.908i)2-s + (0.528 + 0.528i)3-s + (−0.650 − 0.759i)4-s + (1.31 − 1.31i)5-s + (0.701 − 0.259i)6-s + 0.723i·7-s + (−0.962 + 0.272i)8-s − 0.441i·9-s + (−0.643 − 1.74i)10-s + (−0.894 + 0.894i)11-s + (0.0580 − 0.745i)12-s + (0.321 + 0.321i)13-s + (0.657 + 0.302i)14-s + 1.38·15-s + (−0.154 + 0.987i)16-s + 0.242·17-s + ⋯

Functional equation

Λ(s)=(272s/2 ΓC(s) L(s)=((0.235+0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(272s/2 ΓC(s+1/2) L(s)=((0.235+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 272272    =    24⋅172^{4} \cdot 17
Sign: 0.235+0.971i0.235 + 0.971i
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), 0.235+0.971i)(2,\ 272,\ (\ :1/2),\ 0.235 + 0.971i)

Particular Values

L(1)L(1) ≈\approx 1.52651−1.20131i1.52651 - 1.20131i
L(12)L(\frac12) ≈\approx 1.52651−1.20131i1.52651 - 1.20131i
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+(−0.591+1.28i)T 1 + (-0.591 + 1.28i)T
17 1−T 1 - T
good3 1+(−0.915−0.915i)T+3iT2 1 + (-0.915 - 0.915i)T + 3iT^{2}
5 1+(−2.93+2.93i)T−5iT2 1 + (-2.93 + 2.93i)T - 5iT^{2}
7 1−1.91iT−7T2 1 - 1.91iT - 7T^{2}
11 1+(2.96−2.96i)T−11iT2 1 + (2.96 - 2.96i)T - 11iT^{2}
13 1+(−1.16−1.16i)T+13iT2 1 + (-1.16 - 1.16i)T + 13iT^{2}
19 1+(1.64+1.64i)T+19iT2 1 + (1.64 + 1.64i)T + 19iT^{2}
23 1−0.425iT−23T2 1 - 0.425iT - 23T^{2}
29 1+(−4.17−4.17i)T+29iT2 1 + (-4.17 - 4.17i)T + 29iT^{2}
31 1+4.08T+31T2 1 + 4.08T + 31T^{2}
37 1+(7.58−7.58i)T−37iT2 1 + (7.58 - 7.58i)T - 37iT^{2}
41 1−11.1iT−41T2 1 - 11.1iT - 41T^{2}
43 1+(−4.70+4.70i)T−43iT2 1 + (-4.70 + 4.70i)T - 43iT^{2}
47 1+1.47T+47T2 1 + 1.47T + 47T^{2}
53 1+(1.05−1.05i)T−53iT2 1 + (1.05 - 1.05i)T - 53iT^{2}
59 1+(6.30−6.30i)T−59iT2 1 + (6.30 - 6.30i)T - 59iT^{2}
61 1+(6.38+6.38i)T+61iT2 1 + (6.38 + 6.38i)T + 61iT^{2}
67 1+(3.95+3.95i)T+67iT2 1 + (3.95 + 3.95i)T + 67iT^{2}
71 1−7.48iT−71T2 1 - 7.48iT - 71T^{2}
73 1+0.137iT−73T2 1 + 0.137iT - 73T^{2}
79 1−15.5T+79T2 1 - 15.5T + 79T^{2}
83 1+(9.36+9.36i)T+83iT2 1 + (9.36 + 9.36i)T + 83iT^{2}
89 1+3.78iT−89T2 1 + 3.78iT - 89T^{2}
97 1−1.68T+97T2 1 - 1.68T + 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

−12.07179637516697618454660965825, −10.50755738080881482107982012550, −9.765115394238596899831792856472, −9.098438697875161510389873565639, −8.513656655871312153503822869288, −6.26592179240806975059960045285, −5.21153173583690424277309430698, −4.52423594139995495860399367821, −2.85769243175883387662621961928, −1.64906091974799633536123633413, 2.41642604888447670163514799311, 3.50136724210261868124818845508, 5.38253798240969640712972996516, 6.18303671762093272873386909491, 7.19935457864013780510132696302, 7.87687663190405069411571331282, 9.012475511911690570924249019470, 10.37223938221981077019423101694, 10.82749002217119166598941826999, 12.63851991621306202024319815616

Graph of the ZZ-function along the critical line