↑

Properties

Label 1-51-51.47-r1-0-0
Degree $1$
Conductor $51$
Sign $0.788 - 0.615i$
Analytic cond. $5.48071$
Root an. cond. $5.48071$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − i·5-s − i·7-s + 8-s − i·10-s + i·11-s + 13-s − i·14-s + 16-s − 19-s − i·20-s + i·22-s + i·23-s − 25-s + 26-s + ⋯
L(s)  = 1  + 2-s + 4-s − i·5-s − i·7-s + 8-s − i·10-s + i·11-s + 13-s − i·14-s + 16-s − 19-s − i·20-s + i·22-s + i·23-s − 25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(5.48071\)
Root analytic conductor: \(5.48071\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (1:\ ),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.672958108 - 0.9199004879i\)
\(L(\frac12)\) \(\approx\) \(2.672958108 - 0.9199004879i\)
\(L(1)\) \(\approx\) \(1.928414006 - 0.3894258837i\)
\(L(1)\) \(\approx\) \(1.928414006 - 0.3894258837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
13 \( 1 - iT \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 \)
31 \( 1 - iT \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 - iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 \)
83 \( 1 + iT \)
89 \( 1 + iT \)
97 \( 1 \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.38356476329067642884693206758, −32.14927472100879125688762779413, −31.1534613843036460580488209308, −30.21113153291036467918273362675, −29.21129016915208050062693979900, −27.886087781047407570914710576367, −26.2160291616873477317017591118, −25.2610778252731973663997274707, −24.03747709746982733958881779342, −22.829811128011679122239007994697, −21.87739836701472278643370414527, −21.02439388900031901179724323399, −19.30269313911077809744978060018, −18.34024529491632444727335038199, −16.38284901242448448680710523128, −15.238200787496468529247526240717, −14.28271734391226090355093325875, −12.98976349004606714125930956926, −11.54129339458912058185009337258, −10.62523179985428436468825371641, −8.45681550538793811029420920375, −6.62713954285006385888213874461, −5.68218661108062921441775016790, −3.68846410743393866349972257376, −2.37659152792282346858494953666, 1.528753327978942981209348598759, 3.839325316752567455540691704026, 4.91554190840436925668697475861, 6.57261454046174921958129783462, 8.06637083969625612212732689440, 10.036386355670647406790283868616, 11.51163076743198454154136990214, 12.87198144277811000666930132611, 13.63831019994323071538439819604, 15.1793307920529668042419059898, 16.37201651221059475463234136141, 17.42591693534099532530297614429, 19.635677134669205131903228781850, 20.50170187630443046232750720096, 21.374684212506118429690421218209, 23.13865760431996882617606264597, 23.56189485218526766363351210505, 24.95839522664079930513399025200, 25.91665583620737115424096680749, 27.74549406695432035446513733803, 28.79218051579949608055500069202, 29.983284995421020561924176345444, 30.95960377648930974114693113450, 32.17275477310682267532508632148, 32.99791896447882294104474580319

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