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Properties

Label 1-2e9-512.333-r0-0-0
Degree $1$
Conductor $512$
Sign $0.698 - 0.715i$
Analytic cond. $2.37771$
Root an. cond. $2.37771$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(2.37771\)
Root analytic conductor: \(2.37771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 512,\ (0:\ ),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.022453139 - 0.8523035194i\)
\(L(\frac12)\) \(\approx\) \(2.022453139 - 0.8523035194i\)
\(L(1)\) \(\approx\) \(1.576501817 - 0.3764925100i\)
\(L(1)\) \(\approx\) \(1.576501817 - 0.3764925100i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.989 - 0.146i)T \)
5 \( 1 + (-0.0490 - 0.998i)T \)
7 \( 1 + (0.881 - 0.471i)T \)
11 \( 1 + (0.857 + 0.514i)T \)
13 \( 1 + (-0.740 + 0.671i)T \)
17 \( 1 + (0.195 - 0.980i)T \)
19 \( 1 + (0.903 + 0.427i)T \)
23 \( 1 + (-0.773 + 0.634i)T \)
29 \( 1 + (-0.242 + 0.970i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (-0.941 - 0.336i)T \)
41 \( 1 + (-0.995 - 0.0980i)T \)
43 \( 1 + (-0.146 + 0.989i)T \)
47 \( 1 + (-0.555 - 0.831i)T \)
53 \( 1 + (-0.242 - 0.970i)T \)
59 \( 1 + (0.740 + 0.671i)T \)
61 \( 1 + (-0.803 - 0.595i)T \)
67 \( 1 + (-0.595 + 0.803i)T \)
71 \( 1 + (0.290 - 0.956i)T \)
73 \( 1 + (0.881 + 0.471i)T \)
79 \( 1 + (-0.831 - 0.555i)T \)
83 \( 1 + (-0.941 + 0.336i)T \)
89 \( 1 + (-0.773 - 0.634i)T \)
97 \( 1 + (0.382 + 0.923i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line