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Properties

Label 2-546-13.10-c1-0-7
Degree 22
Conductor 546546
Sign 0.265+0.964i0.265 + 0.964i
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 0.732i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (4.5 − 2.59i)11-s − 0.999·12-s + (0.866 − 3.5i)13-s + 0.999·14-s + (0.633 − 0.366i)15-s + (−0.5 − 0.866i)16-s + (−1.13 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.327i·5-s + (−0.353 − 0.204i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (1.35 − 0.783i)11-s − 0.288·12-s + (0.240 − 0.970i)13-s + 0.267·14-s + (0.163 − 0.0945i)15-s + (−0.125 − 0.216i)16-s + (−0.275 + 0.476i)17-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((0.265+0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((0.265+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 0.265+0.964i0.265 + 0.964i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(127,⋅)\chi_{546} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 0.265+0.964i)(2,\ 546,\ (\ :1/2),\ 0.265 + 0.964i)

Particular Values

L(1)L(1) ≈\approx 1.58510−1.20823i1.58510 - 1.20823i
L(12)L(\frac12) ≈\approx 1.58510−1.20823i1.58510 - 1.20823i
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−0.866+3.5i)T 1 + (-0.866 + 3.5i)T
good5 1−0.732iT−5T2 1 - 0.732iT - 5T^{2}
11 1+(−4.5+2.59i)T+(5.5−9.52i)T2 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2}
17 1+(1.13−1.96i)T+(−8.5−14.7i)T2 1 + (1.13 - 1.96i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.401+0.232i)T+(9.5+16.4i)T2 1 + (0.401 + 0.232i)T + (9.5 + 16.4i)T^{2}
23 1+(3.73+6.46i)T+(−11.5+19.9i)T2 1 + (3.73 + 6.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.76−3.06i)T+(−14.5+25.1i)T2 1 + (-1.76 - 3.06i)T + (-14.5 + 25.1i)T^{2}
31 1+3.26iT−31T2 1 + 3.26iT - 31T^{2}
37 1+(−5.83+3.36i)T+(18.5−32.0i)T2 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2}
41 1+(7.33−4.23i)T+(20.5−35.5i)T2 1 + (7.33 - 4.23i)T + (20.5 - 35.5i)T^{2}
43 1+(4.36−7.56i)T+(−21.5−37.2i)T2 1 + (4.36 - 7.56i)T + (-21.5 - 37.2i)T^{2}
47 1−3.92iT−47T2 1 - 3.92iT - 47T^{2}
53 1+9.92T+53T2 1 + 9.92T + 53T^{2}
59 1+(−7.73−4.46i)T+(29.5+51.0i)T2 1 + (-7.73 - 4.46i)T + (29.5 + 51.0i)T^{2}
61 1+(1.86−3.23i)T+(−30.5−52.8i)T2 1 + (1.86 - 3.23i)T + (-30.5 - 52.8i)T^{2}
67 1+(−8.66+5i)T+(33.5−58.0i)T2 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2}
71 1+(−6.29−3.63i)T+(35.5+61.4i)T2 1 + (-6.29 - 3.63i)T + (35.5 + 61.4i)T^{2}
73 1+1.46iT−73T2 1 + 1.46iT - 73T^{2}
79 1−9T+79T2 1 - 9T + 79T^{2}
83 1−9.26iT−83T2 1 - 9.26iT - 83T^{2}
89 1+(14.5−8.42i)T+(44.5−77.0i)T2 1 + (14.5 - 8.42i)T + (44.5 - 77.0i)T^{2}
97 1+(12.2+7.09i)T+(48.5+84.0i)T2 1 + (12.2 + 7.09i)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

−10.98818831520048276307149138032, −9.993870318793570289961264069648, −8.743238234612674527219204522098, −7.967288604788083818557224881256, −6.52543997429274100138818482948, −6.21537209994820167311612232207, −4.97470069741193896304403926685, −3.80530960900091386705203995356, −2.63938249488714978254864761019, −1.14369903608284725194696881675, 1.74365207528250797515571452263, 3.62903464659131082213049936290, 4.41470713949315019862199537618, 5.21206160125379937896494216959, 6.46706840243440355843281942084, 7.06230048433254122061983068154, 8.363042712629729854518786948261, 9.251461091978430513803618732640, 10.00927162009268463229537775197, 11.34316495980674868734778896986

Graph of the ZZ-function along the critical line