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Properties

Label 2-546-13.12-c1-0-5
Degree 22
Conductor 546546
Sign 0.832−0.554i0.832 - 0.554i
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  + i·2-s − 3-s − 4-s − i·5-s − i·6-s + i·7-s − i·8-s + 9-s + 10-s − i·11-s + 12-s + (3 − 2i)13-s − 14-s + i·15-s + 16-s + 17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.301i·11-s + 0.288·12-s + (0.832 − 0.554i)13-s − 0.267·14-s + 0.258i·15-s + 0.250·16-s + 0.242·17-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((0.832−0.554i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((0.832−0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 0.832−0.554i0.832 - 0.554i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(337,⋅)\chi_{546} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 0.832−0.554i)(2,\ 546,\ (\ :1/2),\ 0.832 - 0.554i)

Particular Values

L(1)L(1) ≈\approx 1.13638+0.344069i1.13638 + 0.344069i
L(12)L(\frac12) ≈\approx 1.13638+0.344069i1.13638 + 0.344069i
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−iT 1 - iT
3 1+T 1 + T
7 1−iT 1 - iT
13 1+(−3+2i)T 1 + (-3 + 2i)T
good5 1+iT−5T2 1 + iT - 5T^{2}
11 1+iT−11T2 1 + iT - 11T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1−iT−19T2 1 - iT - 19T^{2}
23 1−3T+23T2 1 - 3T + 23T^{2}
29 1−9T+29T2 1 - 9T + 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1−9iT−37T2 1 - 9iT - 37T^{2}
41 1+8iT−41T2 1 + 8iT - 41T^{2}
43 1−7T+43T2 1 - 7T + 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1−11T+61T2 1 - 11T + 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+6iT−71T2 1 + 6iT - 71T^{2}
73 1+11iT−73T2 1 + 11iT - 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+6iT−83T2 1 + 6iT - 83T^{2}
89 1+12iT−89T2 1 + 12iT - 89T^{2}
97 1−2iT−97T2 1 - 2iT - 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

−10.78104301170401505307511686094, −10.03956400679909049827172074353, −8.783531309560911449799472382690, −8.383785026984063472299012064678, −7.13626991888380856938724060512, −6.23283583615769451213451930779, −5.42033514195581625818776081159, −4.58595412521107281643708288977, −3.18728483142858966879703378545, −1.03407213013071041944117832355, 1.11881376874371552172747247000, 2.72104849752177574724365641659, 3.98271519982292327097434245597, 4.89211644729518278801671489593, 6.16423625528239770109262574331, 6.98106462795789150955807148679, 8.109639533376156155836635696537, 9.219480901588857273058422341604, 10.02004710064398676615273167938, 11.04505362659265251242170856084

Graph of the ZZ-function along the critical line