↑

Properties

Label 1-51-51.38-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: $\chi_{51} (38, \cdot )$
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

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

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