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Properties

Label 1-2e9-512.333-r0-0-0
Degree 11
Conductor 512512
Sign 0.698−0.715i0.698 - 0.715i
Analytic cond. 2.377712.37771
Root an. cond. 2.377712.37771
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.989 − 0.146i)3-s + (−0.0490 − 0.998i)5-s + (0.881 − 0.471i)7-s + (0.956 − 0.290i)9-s + (0.857 + 0.514i)11-s + (−0.740 + 0.671i)13-s + (−0.195 − 0.980i)15-s + (0.195 − 0.980i)17-s + (0.903 + 0.427i)19-s + (0.803 − 0.595i)21-s + (−0.773 + 0.634i)23-s + (−0.995 + 0.0980i)25-s + (0.903 − 0.427i)27-s + (−0.242 + 0.970i)29-s + (0.923 − 0.382i)31-s + ⋯
L(s)  = 1  + (0.989 − 0.146i)3-s + (−0.0490 − 0.998i)5-s + (0.881 − 0.471i)7-s + (0.956 − 0.290i)9-s + (0.857 + 0.514i)11-s + (−0.740 + 0.671i)13-s + (−0.195 − 0.980i)15-s + (0.195 − 0.980i)17-s + (0.903 + 0.427i)19-s + (0.803 − 0.595i)21-s + (−0.773 + 0.634i)23-s + (−0.995 + 0.0980i)25-s + (0.903 − 0.427i)27-s + (−0.242 + 0.970i)29-s + (0.923 − 0.382i)31-s + ⋯

Functional equation

Λ(s)=(512s/2 ΓR(s) L(s)=((0.698−0.715i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(512s/2 ΓR(s) L(s)=((0.698−0.715i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 512512    =    292^{9}
Sign: 0.698−0.715i0.698 - 0.715i
Analytic conductor: 2.377712.37771
Root analytic conductor: 2.377712.37771
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ512(333,⋅)\chi_{512} (333, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 512, (0: ), 0.698−0.715i)(1,\ 512,\ (0:\ ),\ 0.698 - 0.715i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.022453139−0.8523035194i2.022453139 - 0.8523035194i
L(12)L(\frac12) ≈\approx 2.022453139−0.8523035194i2.022453139 - 0.8523035194i
L(1)L(1) ≈\approx 1.576501817−0.3764925100i1.576501817 - 0.3764925100i
L(1)L(1) ≈\approx 1.576501817−0.3764925100i1.576501817 - 0.3764925100i

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
good3 1+(0.989−0.146i)T 1 + (0.989 - 0.146i)T
5 1+(−0.0490−0.998i)T 1 + (-0.0490 - 0.998i)T
7 1+(0.881−0.471i)T 1 + (0.881 - 0.471i)T
11 1+(0.857+0.514i)T 1 + (0.857 + 0.514i)T
13 1+(−0.740+0.671i)T 1 + (-0.740 + 0.671i)T
17 1+(0.195−0.980i)T 1 + (0.195 - 0.980i)T
19 1+(0.903+0.427i)T 1 + (0.903 + 0.427i)T
23 1+(−0.773+0.634i)T 1 + (-0.773 + 0.634i)T
29 1+(−0.242+0.970i)T 1 + (-0.242 + 0.970i)T
31 1+(0.923−0.382i)T 1 + (0.923 - 0.382i)T
37 1+(−0.941−0.336i)T 1 + (-0.941 - 0.336i)T
41 1+(−0.995−0.0980i)T 1 + (-0.995 - 0.0980i)T
43 1+(−0.146+0.989i)T 1 + (-0.146 + 0.989i)T
47 1+(−0.555−0.831i)T 1 + (-0.555 - 0.831i)T
53 1+(−0.242−0.970i)T 1 + (-0.242 - 0.970i)T
59 1+(0.740+0.671i)T 1 + (0.740 + 0.671i)T
61 1+(−0.803−0.595i)T 1 + (-0.803 - 0.595i)T
67 1+(−0.595+0.803i)T 1 + (-0.595 + 0.803i)T
71 1+(0.290−0.956i)T 1 + (0.290 - 0.956i)T
73 1+(0.881+0.471i)T 1 + (0.881 + 0.471i)T
79 1+(−0.831−0.555i)T 1 + (-0.831 - 0.555i)T
83 1+(−0.941+0.336i)T 1 + (-0.941 + 0.336i)T
89 1+(−0.773−0.634i)T 1 + (-0.773 - 0.634i)T
97 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−24.023213628327943814522402961127, −22.51496225687797513023050975246, −21.99135063347003286510923058945, −21.24327275916148965993606576358, −20.25269071590676511802292505126, −19.429071800018639139413676494774, −18.78495012074776958121681960829, −17.8912855488191879635882447044, −17.03087045888736954964473691496, −15.53294758891110098352702661129, −15.13975006615696371420471665374, −14.20991739660068177796589746891, −13.835463939390175626998694794370, −12.388436453071310483473156794202, −11.54558308516554091255051327225, −10.469038526466960758864578893052, −9.74779228917959450718580623847, −8.55250901481680085644252196191, −7.939654778427175459217656973194, −6.967430037328005153038903413910, −5.84148850975083564527031169427, −4.54810887543381053334338058961, −3.47920371736253739074087333857, −2.6311715743682989484108175447, −1.59985902077914957826334679646, 1.2730232976731595924345294049, 1.95369902366472752584108303353, 3.51335597293070673672202403755, 4.44846688526493202765149911213, 5.18329387229340117698339813260, 6.90894884645403451195785737358, 7.62597422183657819264831973329, 8.48394451226662007909061609063, 9.420834979349082347363155422973, 9.968894838420983011991381594310, 11.72410210128083388189233229103, 12.08222140467123598767496452353, 13.35260646942411395707884956090, 14.08423152524271310449765676280, 14.6569717825500291811931732890, 15.802807634882468440066917086313, 16.65177489525246687499258409961, 17.54826072852409836174418300136, 18.41144782007684510368649325392, 19.63838066819613945730544482901, 20.06418057502147827654410009411, 20.80367691916308464646257596963, 21.48122198834723472215291646167, 22.655942310462025698213237567907, 23.84857699779634769907576092837

Graph of the ZZ-function along the critical line