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Properties

Label 1-2e9-512.325-r0-0-0
Degree $1$
Conductor $512$
Sign $0.715 - 0.698i$
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  + (0.803 − 0.595i)3-s + (0.740 − 0.671i)5-s + (−0.471 + 0.881i)7-s + (0.290 − 0.956i)9-s + (0.970 + 0.242i)11-s + (0.998 − 0.0490i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.336 + 0.941i)19-s + (0.146 + 0.989i)21-s + (−0.634 + 0.773i)23-s + (0.0980 − 0.995i)25-s + (−0.336 − 0.941i)27-s + (−0.857 − 0.514i)29-s + (0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)3-s + (0.740 − 0.671i)5-s + (−0.471 + 0.881i)7-s + (0.290 − 0.956i)9-s + (0.970 + 0.242i)11-s + (0.998 − 0.0490i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.336 + 0.941i)19-s + (0.146 + 0.989i)21-s + (−0.634 + 0.773i)23-s + (0.0980 − 0.995i)25-s + (−0.336 − 0.941i)27-s + (−0.857 − 0.514i)29-s + (0.923 + 0.382i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(2.37771\)
Root analytic conductor: \(2.37771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 512,\ (0:\ ),\ 0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952031111 - 0.7945606306i\)
\(L(\frac12)\) \(\approx\) \(1.952031111 - 0.7945606306i\)
\(L(1)\) \(\approx\) \(1.520638985 - 0.3769512766i\)
\(L(1)\) \(\approx\) \(1.520638985 - 0.3769512766i\)

Euler product

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

−23.710894488675774570722624609244, −22.49389961290255045198718931810, −22.077105976399361850472812886044, −21.137387421160799684965127336525, −20.36367920987201069115897037914, −19.50000548400737722632400643365, −18.86659551649904433075815610021, −17.64595256129727040522727982587, −16.855871140243039820048954780620, −15.992923382965814505790816337, −14.97554107512792752543384030623, −14.20930918948399662794117643138, −13.59483170808698545154740043871, −12.81575588702527007132782312818, −11.01675081913047942286305037345, −10.6804765573049366947889978822, −9.60220325844854569284511363833, −8.968871882662774147304718406945, −7.84391209418079589821920856909, −6.671283589126808655620955788936, −6.01806834152786528194285766686, −4.35048596379107347157051456292, −3.70072151433654722343733063833, −2.67642742786918101917185417992, −1.46987439019017598702920268189, 1.2520942139772787537531068619, 2.09813837963228003000494541019, 3.22779915796836636575614619514, 4.359018292145722649081301618489, 5.90113863749781679793902566641, 6.31570265571217896612673272417, 7.66314571071652933108852776373, 8.72294787063540716765890027008, 9.231601651061124164617223081008, 9.98930374631863561870948851231, 11.71558224018077010432451206406, 12.29637488442720004873869634633, 13.306397887647039251430384828428, 13.80956536146090803775389120687, 14.82768402637411303572319277015, 15.74798172562540260517517380440, 16.66818344226364484411952245290, 17.764857806720726613750101906493, 18.4309994835945472216012202211, 19.24200158420523177610875836732, 20.19570681781799109767000437597, 20.805540971692946044279078094301, 21.6674760293497858304525119007, 22.62317040208497718858316056868, 23.62707692095969159257825009543

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