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Properties

Label 1-2e9-512.245-r0-0-0
Degree $1$
Conductor $512$
Sign $0.953 + 0.302i$
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  + (−0.0490 − 0.998i)3-s + (0.514 + 0.857i)5-s + (0.634 + 0.773i)7-s + (−0.995 + 0.0980i)9-s + (0.941 − 0.336i)11-s + (0.242 + 0.970i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (0.146 − 0.989i)19-s + (0.740 − 0.671i)21-s + (−0.290 + 0.956i)23-s + (−0.471 + 0.881i)25-s + (0.146 + 0.989i)27-s + (0.427 + 0.903i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.0490 − 0.998i)3-s + (0.514 + 0.857i)5-s + (0.634 + 0.773i)7-s + (−0.995 + 0.0980i)9-s + (0.941 − 0.336i)11-s + (0.242 + 0.970i)13-s + (0.831 − 0.555i)15-s + (−0.831 − 0.555i)17-s + (0.146 − 0.989i)19-s + (0.740 − 0.671i)21-s + (−0.290 + 0.956i)23-s + (−0.471 + 0.881i)25-s + (0.146 + 0.989i)27-s + (0.427 + 0.903i)29-s + (−0.382 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.953 + 0.302i$
Analytic conductor: \(2.37771\)
Root analytic conductor: \(2.37771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 512,\ (0:\ ),\ 0.953 + 0.302i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530829861 + 0.2366857605i\)
\(L(\frac12)\) \(\approx\) \(1.530829861 + 0.2366857605i\)
\(L(1)\) \(\approx\) \(1.218845819 + 0.005035299893i\)
\(L(1)\) \(\approx\) \(1.218845819 + 0.005035299893i\)

Euler product

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

−23.400060768901990847914242631136, −22.629471432043311488705468427634, −21.78363597289115771593291666145, −20.92516027972586039061982698262, −20.2198723324637560993018779716, −19.89868259040728190572909719471, −18.19278626522944231668752868722, −17.21466230185933361766022929615, −16.951817495782158936364231496872, −15.96408214114949230013754065496, −14.94188965154665752692704714619, −14.23897172266301992305461631469, −13.29238942899257562179307841560, −12.25667946156247018307894599656, −11.228123139975478206616664837764, −10.33239020559081348223531983323, −9.65070106992332957304313285183, −8.62151620533134204772875191064, −7.931021629750403635640637670706, −6.32085323878049252279779442390, −5.49468265018190092683157301734, −4.332806081156152388540296837815, −3.97845667555580937062240757306, −2.270579838064357598817104524082, −0.92723806276324261295685379671, 1.439749116307085072511916291220, 2.244248272348536917661864665919, 3.264607724906605612426606248653, 4.86352463805299283634888286453, 6.00245319961642583326545623140, 6.6881794698920958107892932437, 7.45429948201600306964090752424, 8.825473819571685455538993550015, 9.254796835510365924065295365615, 10.980322613423635114501607887869, 11.441840025918725379257659726424, 12.24961207826437262514021878096, 13.56492499903197363372162752997, 14.032854157724690267472272301595, 14.80525777408425338632790944229, 15.91367095381508676309297805097, 17.20135050582826908948903436348, 17.85594156777888818884957003582, 18.43589382398637409085495844289, 19.27139068990553341829736336613, 20.00154370561318351448894369156, 21.37971552705614259102134916600, 21.95702339550362205653638439958, 22.69978114095606860568299827189, 23.92117699815791440211154893119

Graph of the $Z$-function along the critical line