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Properties

Label 2-546-13.3-c1-0-11
Degree 22
Conductor 546546
Sign −0.597+0.802i-0.597 + 0.802i
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2.56·5-s + (0.499 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.28 + 2.21i)10-s + (−0.780 − 1.35i)11-s − 0.999·12-s + (−0.5 − 3.57i)13-s + 0.999·14-s + (−1.28 − 2.21i)15-s + (−0.5 − 0.866i)16-s + (4.06 − 7.03i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 1.14·5-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.405 + 0.701i)10-s + (−0.235 − 0.407i)11-s − 0.288·12-s + (−0.138 − 0.990i)13-s + 0.267·14-s + (−0.330 − 0.572i)15-s + (−0.125 − 0.216i)16-s + (0.985 − 1.70i)17-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((−0.597+0.802i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((−0.597+0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: −0.597+0.802i-0.597 + 0.802i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(211,⋅)\chi_{546} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), −0.597+0.802i)(2,\ 546,\ (\ :1/2),\ -0.597 + 0.802i)

Particular Values

L(1)L(1) ≈\approx 0.262767−0.523153i0.262767 - 0.523153i
L(12)L(\frac12) ≈\approx 0.262767−0.523153i0.262767 - 0.523153i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1+(0.5+3.57i)T 1 + (0.5 + 3.57i)T
good5 1+2.56T+5T2 1 + 2.56T + 5T^{2}
11 1+(0.780+1.35i)T+(−5.5+9.52i)T2 1 + (0.780 + 1.35i)T + (-5.5 + 9.52i)T^{2}
17 1+(−4.06+7.03i)T+(−8.5−14.7i)T2 1 + (-4.06 + 7.03i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.780−1.35i)T+(−9.5−16.4i)T2 1 + (0.780 - 1.35i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.56+6.16i)T+(−11.5+19.9i)T2 1 + (3.56 + 6.16i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.06+1.83i)T+(−14.5+25.1i)T2 1 + (1.06 + 1.83i)T + (-14.5 + 25.1i)T^{2}
31 1+31T2 1 + 31T^{2}
37 1+(−3.28−5.68i)T+(−18.5+32.0i)T2 1 + (-3.28 - 5.68i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.62+4.54i)T+(−20.5+35.5i)T2 1 + (2.62 + 4.54i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4+6.92i)T+(−21.5−37.2i)T2 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2}
47 1+12.6T+47T2 1 + 12.6T + 47T^{2}
53 1−7T+53T2 1 - 7T + 53T^{2}
59 1+(−1.56+2.70i)T+(−29.5−51.0i)T2 1 + (-1.56 + 2.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.62−4.54i)T+(−30.5−52.8i)T2 1 + (2.62 - 4.54i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.438+0.759i)T+(−33.5+58.0i)T2 1 + (0.438 + 0.759i)T + (-33.5 + 58.0i)T^{2}
71 1+(6.68−11.5i)T+(−35.5−61.4i)T2 1 + (6.68 - 11.5i)T + (-35.5 - 61.4i)T^{2}
73 1+6.56T+73T2 1 + 6.56T + 73T^{2}
79 1+2.43T+79T2 1 + 2.43T + 79T^{2}
83 1−3.12T+83T2 1 - 3.12T + 83T^{2}
89 1+(3.78+6.54i)T+(−44.5+77.0i)T2 1 + (3.78 + 6.54i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.56+7.90i)T+(−48.5−84.0i)T2 1 + (-4.56 + 7.90i)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

−10.34836773062637488730080989998, −9.826142664958310189611096007354, −8.649012574907809725469065691998, −8.071875507341444345947254388877, −7.25387520366286420394293603740, −5.64475336053364533715147800330, −4.55000271140865864625408407922, −3.46046007163665308964432715682, −2.68469348378206225240669971916, −0.37352115187422595505984512029, 1.62359781957480983878608191498, 3.53011459871439766322861903628, 4.39369288985163412354208430824, 5.84798205387652453239356846185, 6.81739302926739690352145119055, 7.73248284358295560912386242723, 8.058218068807758135724303121222, 9.202532832951306140696282577089, 10.06411124389661206173449904135, 11.14387192093863938495354857316

Graph of the ZZ-function along the critical line