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Properties

Label 1-203-203.31-r0-0-0
Degree $1$
Conductor $203$
Sign $0.0423 + 0.999i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (−0.680 + 0.733i)2-s + (−0.563 + 0.826i)3-s + (−0.0747 − 0.997i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (0.781 + 0.623i)8-s + (−0.365 − 0.930i)9-s + (0.997 − 0.0747i)10-s + (−0.930 − 0.365i)11-s + (0.866 + 0.5i)12-s + (0.623 + 0.781i)13-s + (0.974 − 0.222i)15-s + (−0.988 + 0.149i)16-s + (0.866 − 0.5i)17-s + (0.930 + 0.365i)18-s + (0.563 + 0.826i)19-s + ⋯
L(s)  = 1  + (−0.680 + 0.733i)2-s + (−0.563 + 0.826i)3-s + (−0.0747 − 0.997i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (0.781 + 0.623i)8-s + (−0.365 − 0.930i)9-s + (0.997 − 0.0747i)10-s + (−0.930 − 0.365i)11-s + (0.866 + 0.5i)12-s + (0.623 + 0.781i)13-s + (0.974 − 0.222i)15-s + (−0.988 + 0.149i)16-s + (0.866 − 0.5i)17-s + (0.930 + 0.365i)18-s + (0.563 + 0.826i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0423 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0423 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.0423 + 0.999i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.0423 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3880002437 + 0.3719041506i\)
\(L(\frac12)\) \(\approx\) \(0.3880002437 + 0.3719041506i\)
\(L(1)\) \(\approx\) \(0.5102811404 + 0.2584085592i\)
\(L(1)\) \(\approx\) \(0.5102811404 + 0.2584085592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.680 + 0.733i)T \)
3 \( 1 + (-0.563 + 0.826i)T \)
5 \( 1 + (-0.733 - 0.680i)T \)
11 \( 1 + (-0.930 - 0.365i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.563 + 0.826i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
31 \( 1 + (0.294 + 0.955i)T \)
37 \( 1 + (-0.930 + 0.365i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.974 + 0.222i)T \)
47 \( 1 + (0.149 + 0.988i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.997 + 0.0747i)T \)
67 \( 1 + (0.988 + 0.149i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.680 + 0.733i)T \)
79 \( 1 + (0.930 - 0.365i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.680 - 0.733i)T \)
97 \( 1 + (-0.433 + 0.900i)T \)
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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

−26.65740705447579902709829736706, −25.88259351973925248455640341375, −24.88015756742603357499138645309, −23.4847955914397162880088877762, −22.9184223554537862419166855221, −21.95421980808337648272160737525, −20.69910576107601670328242566200, −19.72266804244742041618675419453, −18.799183484773682268310154213519, −18.27296622474731012041401651987, −17.390266205484486038936579011853, −16.25359226486797347422388857538, −15.17507673996703316745523854989, −13.50416985770772957167005820498, −12.71629879210713223075264935311, −11.72554545013667803908758634408, −10.910037214258048074224131739511, −10.147088750202380576197493461461, −8.37383037666138331702421889734, −7.66708489057965694845969167985, −6.76185191519950489795460408130, −5.15017149530488928515511624428, −3.44577927977569286856903997309, −2.39214661168095009182987261167, −0.740627234192568494588595610926, 1.03599160153799697439565412435, 3.536601102859097110377671532724, 4.92732649188189263447352143085, 5.576714417940391067076601172139, 7.007616661829324341544278896532, 8.22458565977679221282131783935, 9.07705840342014476229269688087, 10.14786423382262177908305513961, 11.1415018248991516298877206113, 12.10668688386601114883357931879, 13.70707794858853771948285619081, 14.911866926849948035569712649418, 16.016648519074484238412512091120, 16.210336176118635100050531290892, 17.20185508433444022144795239950, 18.39498543377002120971164681458, 19.2196334785001556560301012657, 20.58239002687357761073609980116, 21.12326886969890235101692310098, 22.79497260415546142125621564538, 23.38839094116819963551488479713, 24.121163045890064773687249423776, 25.31951437257179091950031996615, 26.37194343066783678337600923631, 27.11399014195125529090754525347

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