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Properties

Label 2-272-16.13-c1-0-18
Degree 22
Conductor 272272
Sign −0.921+0.387i-0.921 + 0.387i
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  + (−1.41 + 0.00186i)2-s + (−0.274 − 0.274i)3-s + (1.99 − 0.00528i)4-s + (−2.33 + 2.33i)5-s + (0.389 + 0.388i)6-s + 0.445i·7-s + (−2.82 + 0.0112i)8-s − 2.84i·9-s + (3.30 − 3.31i)10-s + (−0.541 + 0.541i)11-s + (−0.551 − 0.548i)12-s + (−3.22 − 3.22i)13-s + (−0.000833 − 0.630i)14-s + 1.28·15-s + (3.99 − 0.0211i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00132i)2-s + (−0.158 − 0.158i)3-s + (0.999 − 0.00264i)4-s + (−1.04 + 1.04i)5-s + (0.158 + 0.158i)6-s + 0.168i·7-s + (−0.999 + 0.00396i)8-s − 0.949i·9-s + (1.04 − 1.04i)10-s + (−0.163 + 0.163i)11-s + (−0.159 − 0.158i)12-s + (−0.894 − 0.894i)13-s + (−0.000222 − 0.168i)14-s + 0.332·15-s + (0.999 − 0.00528i)16-s − 0.242·17-s + ⋯

Functional equation

Λ(s)=(272s/2 ΓC(s) L(s)=((−0.921+0.387i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(272s/2 ΓC(s+1/2) L(s)=((−0.921+0.387i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 272272    =    24⋅172^{4} \cdot 17
Sign: −0.921+0.387i-0.921 + 0.387i
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), −0.921+0.387i)(2,\ 272,\ (\ :1/2),\ -0.921 + 0.387i)

Particular Values

L(1)L(1) ≈\approx 0.0153607−0.0761705i0.0153607 - 0.0761705i
L(12)L(\frac12) ≈\approx 0.0153607−0.0761705i0.0153607 - 0.0761705i
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+(1.41−0.00186i)T 1 + (1.41 - 0.00186i)T
17 1+T 1 + T
good3 1+(0.274+0.274i)T+3iT2 1 + (0.274 + 0.274i)T + 3iT^{2}
5 1+(2.33−2.33i)T−5iT2 1 + (2.33 - 2.33i)T - 5iT^{2}
7 1−0.445iT−7T2 1 - 0.445iT - 7T^{2}
11 1+(0.541−0.541i)T−11iT2 1 + (0.541 - 0.541i)T - 11iT^{2}
13 1+(3.22+3.22i)T+13iT2 1 + (3.22 + 3.22i)T + 13iT^{2}
19 1+(5.38+5.38i)T+19iT2 1 + (5.38 + 5.38i)T + 19iT^{2}
23 1−7.35iT−23T2 1 - 7.35iT - 23T^{2}
29 1+(4.91+4.91i)T+29iT2 1 + (4.91 + 4.91i)T + 29iT^{2}
31 1+4.01T+31T2 1 + 4.01T + 31T^{2}
37 1+(2.25−2.25i)T−37iT2 1 + (2.25 - 2.25i)T - 37iT^{2}
41 1+8.61iT−41T2 1 + 8.61iT - 41T^{2}
43 1+(2.61−2.61i)T−43iT2 1 + (2.61 - 2.61i)T - 43iT^{2}
47 1−0.997T+47T2 1 - 0.997T + 47T^{2}
53 1+(−2.69+2.69i)T−53iT2 1 + (-2.69 + 2.69i)T - 53iT^{2}
59 1+(8.74−8.74i)T−59iT2 1 + (8.74 - 8.74i)T - 59iT^{2}
61 1+(−7.55−7.55i)T+61iT2 1 + (-7.55 - 7.55i)T + 61iT^{2}
67 1+(−9.29−9.29i)T+67iT2 1 + (-9.29 - 9.29i)T + 67iT^{2}
71 1−2.68iT−71T2 1 - 2.68iT - 71T^{2}
73 1+0.113iT−73T2 1 + 0.113iT - 73T^{2}
79 1−5.48T+79T2 1 - 5.48T + 79T^{2}
83 1+(4.04+4.04i)T+83iT2 1 + (4.04 + 4.04i)T + 83iT^{2}
89 1+12.6iT−89T2 1 + 12.6iT - 89T^{2}
97 1+14.5T+97T2 1 + 14.5T + 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

−11.38958819193385792596498468933, −10.60931723160311279301585374933, −9.615674645682939325007375518855, −8.586194207767826482321417477616, −7.36686082414033697954587902901, −7.05399553495596153440946594795, −5.74613058997108759041601582552, −3.73778277275632912080502894884, −2.55240867109139727274775062741, −0.07742693805914477655838073514, 2.01849518020895850763191125338, 4.03870989355796850754557465009, 5.14288923709050637717257087586, 6.69162577652936359591783991086, 7.82227039607648332916808033106, 8.391049434796618839839993763398, 9.325687833593365296021937417832, 10.51702423278383171109051843625, 11.16302555986734476910229059812, 12.26066424095010372414034046325

Graph of the ZZ-function along the critical line