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Properties

Label 1-2e9-512.125-r0-0-0
Degree $1$
Conductor $512$
Sign $-0.302 - 0.953i$
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  + (0.427 − 0.903i)3-s + (0.989 + 0.146i)5-s + (−0.0980 − 0.995i)7-s + (−0.634 − 0.773i)9-s + (0.998 − 0.0490i)11-s + (−0.803 − 0.595i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (−0.970 − 0.242i)19-s + (−0.941 − 0.336i)21-s + (−0.471 + 0.881i)23-s + (0.956 + 0.290i)25-s + (−0.970 + 0.242i)27-s + (0.740 − 0.671i)29-s + (0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (0.427 − 0.903i)3-s + (0.989 + 0.146i)5-s + (−0.0980 − 0.995i)7-s + (−0.634 − 0.773i)9-s + (0.998 − 0.0490i)11-s + (−0.803 − 0.595i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (−0.970 − 0.242i)19-s + (−0.941 − 0.336i)21-s + (−0.471 + 0.881i)23-s + (0.956 + 0.290i)25-s + (−0.970 + 0.242i)27-s + (0.740 − 0.671i)29-s + (0.382 + 0.923i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.302 - 0.953i$
Analytic conductor: \(2.37771\)
Root analytic conductor: \(2.37771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 512,\ (0:\ ),\ -0.302 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004971670 - 1.372569813i\)
\(L(\frac12)\) \(\approx\) \(1.004971670 - 1.372569813i\)
\(L(1)\) \(\approx\) \(1.174435627 - 0.6407913774i\)
\(L(1)\) \(\approx\) \(1.174435627 - 0.6407913774i\)

Euler product

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

−24.107540613496022834928757610552, −22.49764382959435232683655324042, −22.00239265273452632433976585708, −21.4724827889337374198251207475, −20.65945655064078326097749521951, −19.62159070131330197409183674028, −18.9721444676795774852916997194, −17.71481001927232259889314122356, −16.95035256467872780192576894848, −16.27516863192200375295769680632, −14.994695910933014606453216174839, −14.65898482886790870637038035814, −13.70989203395122211169645123127, −12.60278159048517137425179290644, −11.72850082599226708126557707497, −10.482833384733538271622140181945, −9.80499188504763249182269879163, −8.87270680616835784431337275780, −8.50478843649648898863403153896, −6.68377672082373113725711487017, −5.91657682619228304523038206790, −4.84164221603122405418926213504, −3.98767405409493199425530866759, −2.54109731253532145269442654813, −1.9310108169926703448197947861, 0.85512621008299191386762486486, 1.989827704646080875748923653220, 2.94747948997248697506228377993, 4.199316188680660672854245671325, 5.5506895267882805545229167512, 6.69641820311182170364001741373, 7.07124446228437158856521442441, 8.29692666024639884533470932804, 9.31115222679207138246756592892, 10.07758379684350422030597810347, 11.18521549869772103692028985591, 12.28719856041952176274116113246, 13.11962036815663320879185831005, 13.991809852604931210704435224575, 14.277818509465705533935258904228, 15.51611725780296323315318785328, 16.98526276487896643677542729162, 17.42786560952100421589582343450, 18.05692590752419400743388321947, 19.38126012005734775803032632166, 19.756418604610038513161087829995, 20.6685696735116203106538544877, 21.6454742395536681132033637194, 22.61772489825173202219255643249, 23.3272725069369910272538102596

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