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Properties

Label 2-273-273.152-c1-0-19
Degree 22
Conductor 273273
Sign 0.924+0.381i0.924 + 0.381i
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  + (−0.667 + 0.385i)2-s + (1.00 − 1.40i)3-s + (−0.702 + 1.21i)4-s + (0.907 − 1.57i)5-s + (−0.130 + 1.32i)6-s + (1.94 + 1.78i)7-s − 2.62i·8-s + (−0.964 − 2.84i)9-s + 1.39i·10-s − 0.582i·11-s + (1.00 + 2.21i)12-s + (3.53 + 0.728i)13-s + (−1.99 − 0.442i)14-s + (−1.29 − 2.86i)15-s + (−0.392 − 0.680i)16-s + (−0.479 + 0.829i)17-s + ⋯
L(s)  = 1  + (−0.472 + 0.272i)2-s + (0.582 − 0.812i)3-s + (−0.351 + 0.608i)4-s + (0.405 − 0.703i)5-s + (−0.0534 + 0.542i)6-s + (0.736 + 0.676i)7-s − 0.928i·8-s + (−0.321 − 0.946i)9-s + 0.442i·10-s − 0.175i·11-s + (0.289 + 0.640i)12-s + (0.979 + 0.201i)13-s + (−0.532 − 0.118i)14-s + (−0.335 − 0.739i)15-s + (−0.0982 − 0.170i)16-s + (−0.116 + 0.201i)17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.924+0.381i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.924+0.381i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.924+0.381i0.924 + 0.381i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(152,⋅)\chi_{273} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.924+0.381i)(2,\ 273,\ (\ :1/2),\ 0.924 + 0.381i)

Particular Values

L(1)L(1) ≈\approx 1.23893−0.245804i1.23893 - 0.245804i
L(12)L(\frac12) ≈\approx 1.23893−0.245804i1.23893 - 0.245804i
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)
bad3 1+(−1.00+1.40i)T 1 + (-1.00 + 1.40i)T
7 1+(−1.94−1.78i)T 1 + (-1.94 - 1.78i)T
13 1+(−3.53−0.728i)T 1 + (-3.53 - 0.728i)T
good2 1+(0.667−0.385i)T+(1−1.73i)T2 1 + (0.667 - 0.385i)T + (1 - 1.73i)T^{2}
5 1+(−0.907+1.57i)T+(−2.5−4.33i)T2 1 + (-0.907 + 1.57i)T + (-2.5 - 4.33i)T^{2}
11 1+0.582iT−11T2 1 + 0.582iT - 11T^{2}
17 1+(0.479−0.829i)T+(−8.5−14.7i)T2 1 + (0.479 - 0.829i)T + (-8.5 - 14.7i)T^{2}
19 1+1.92iT−19T2 1 + 1.92iT - 19T^{2}
23 1+(−7.56+4.36i)T+(11.5−19.9i)T2 1 + (-7.56 + 4.36i)T + (11.5 - 19.9i)T^{2}
29 1+(4.26+2.46i)T+(14.5+25.1i)T2 1 + (4.26 + 2.46i)T + (14.5 + 25.1i)T^{2}
31 1+(2.83−1.63i)T+(15.5−26.8i)T2 1 + (2.83 - 1.63i)T + (15.5 - 26.8i)T^{2}
37 1+(−0.899−1.55i)T+(−18.5+32.0i)T2 1 + (-0.899 - 1.55i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.73−2.99i)T+(−20.5−35.5i)T2 1 + (1.73 - 2.99i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.367+0.636i)T+(−21.5+37.2i)T2 1 + (0.367 + 0.636i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.49−6.04i)T+(−23.5−40.7i)T2 1 + (3.49 - 6.04i)T + (-23.5 - 40.7i)T^{2}
53 1+(5.08−2.93i)T+(26.5−45.8i)T2 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2}
59 1+(3.82−6.63i)T+(−29.5−51.0i)T2 1 + (3.82 - 6.63i)T + (-29.5 - 51.0i)T^{2}
61 1+3.82iT−61T2 1 + 3.82iT - 61T^{2}
67 1+10.1T+67T2 1 + 10.1T + 67T^{2}
71 1+(9.73−5.61i)T+(35.5−61.4i)T2 1 + (9.73 - 5.61i)T + (35.5 - 61.4i)T^{2}
73 1+(7.30−4.21i)T+(36.5−63.2i)T2 1 + (7.30 - 4.21i)T + (36.5 - 63.2i)T^{2}
79 1+(−6.35+11.0i)T+(−39.5−68.4i)T2 1 + (-6.35 + 11.0i)T + (-39.5 - 68.4i)T^{2}
83 1+7.45T+83T2 1 + 7.45T + 83T^{2}
89 1+(−3.53−6.12i)T+(−44.5+77.0i)T2 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2}
97 1+(−7.89+4.55i)T+(48.5−84.0i)T2 1 + (-7.89 + 4.55i)T + (48.5 - 84.0i)T^{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

−12.04334004527806081916628514999, −11.03398019559357400820056911598, −9.254935268324535892143078858510, −8.879832234993674837023570151229, −8.208396182165102195658060637800, −7.18687780586697514311254323271, −6.02057761827621362786964614838, −4.60973647788387696773095062365, −3.04760545010185883509705610455, −1.35852628583970183163368662436, 1.73803409946488639742259150939, 3.37145380624365688802916653994, 4.70638243720170058813576881453, 5.72152385873393635558762176657, 7.27841047933777935161175039345, 8.414886900834145846961739005827, 9.241545126563646949685915696871, 10.13242778402532643362276544386, 10.85770069933285304219087981356, 11.26728406463004315963429472070

Graph of the ZZ-function along the critical line