Properties

Degree 1
Conductor 29
Sign $-0.965 - 0.259i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.974 + 0.222i)2-s + (−0.433 + 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s + i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s i·17-s + ⋯
L(s,χ)  = 1  + (−0.974 + 0.222i)2-s + (−0.433 + 0.900i)3-s + (0.900 − 0.433i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.781 − 0.623i)11-s + i·12-s + (−0.623 + 0.781i)13-s + (0.974 + 0.222i)14-s + (−0.974 − 0.222i)15-s + (0.623 − 0.781i)16-s i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.965 - 0.259i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.965 - 0.259i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.965 - 0.259i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (14, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (1:\ ),\ -0.965 - 0.259i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.03893022907 + 0.2945361561i$
$L(\frac12,\chi)$  $\approx$  $-0.03893022907 + 0.2945361561i$
$L(\chi,1)$  $\approx$  0.3747250616 + 0.2511309087i
$L(1,\chi)$  $\approx$  0.3747250616 + 0.2511309087i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−36.32884897925973751989944978013, −35.2287473100929020355826505757, −34.45580871507419549027526719744, −32.73570361008079025482551567695, −31.0516154965372715801311961486, −29.61490271416393009660347178927, −28.62418795508065674265655458023, −28.05554372004965076718262632396, −26.01858014764433045097600224549, −25.01209183772276175636572737841, −23.95026749090851111165224916931, −22.13517852382507452923648244378, −20.35801547798586845634103174551, −19.34121122079636210907478682492, −17.99049740616847144593320782810, −16.99562651617503798887169364400, −15.6912475455025026781435499883, −12.881093689766006580809061459344, −12.335457616553437763722478223536, −10.384437039107422299995041945529, −8.817102143627068953062667097247, −7.37199325865796530868825297488, −5.69474229033383494236138712733, −2.30697289594641798048655483506, −0.2862934905105097622415294941, 3.11052327318947713514380790737, 5.81118686132454743263652222646, 7.236135164049806406728716369542, 9.45973420770702444627571995304, 10.32166071270391729715886245053, 11.548331018847100495150821730577, 14.22083851511268109240505224622, 15.747003021916387875952157815752, 16.62969732897592530304161592763, 18.08164008228776669357575914508, 19.32358314422320525374585276070, 20.898215721725029759565808195954, 22.278173584955518879505979603704, 23.605109310255890352744603238731, 25.59789200548448145166254240641, 26.52134535464006100913263987891, 27.226945801886639338215726681246, 29.05268673290717849849720695740, 29.38735919294798782692708859755, 31.71972518357945938458926289390, 33.262254115807369595485829812974, 33.943576809860669899419047960466, 35.008586385972894814041117190877, 36.46960502124592209044403414400, 37.80993381257266007001975821627

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