Properties

Degree 1
Conductor 29
Sign $0.715 - 0.698i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.715 - 0.698i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.715 - 0.698i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.715 - 0.698i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (10, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (1:\ ),\ 0.715 - 0.698i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.019526541 - 0.8219827192i$
$L(\frac12,\chi)$  $\approx$  $2.019526541 - 0.8219827192i$
$L(\chi,1)$  $\approx$  1.607775250 - 0.5051594150i
$L(1,\chi)$  $\approx$  1.607775250 - 0.5051594150i

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

−36.62206442634949235245163788668, −36.14000881168350163576523553163, −34.50000054051843011200847399237, −33.478261346335069698301720203178, −31.98318786162667095218734398665, −31.295303298256065065819407533168, −30.01747036729571956083038243042, −28.408289135711714958968182789596, −26.47143978210190184320581019194, −25.469005199037844790533722915197, −24.57250497752782310610425371287, −23.616881345228886736821371650429, −21.61352561586080626114205134354, −20.76948549474588535844032025366, −18.52788973054692222833966804559, −17.68173701727030892970621120911, −15.89040691336259250312144215255, −14.44829687946804026573966876336, −13.48026980732169470106766841378, −12.25367715325095036270516538500, −9.264223853844620541376941805765, −8.21062628537667045847387786286, −6.53123332195502238356460766533, −4.90641606986875696017892434199, −2.37695957689145852606659833082, 2.105384994079940654353416695618, 3.75069900661620720258363749484, 5.490326561383460650308063631744, 8.25947504168891974121934927231, 10.17015652356203222584299064695, 10.657797171874414295373478706463, 13.12162943458100624006570019854, 14.05246429032412334651953518349, 15.2142241317931708151866077725, 17.48767435345017586633011991631, 18.96065385844321962134594341719, 20.49637552452733423484022592001, 21.14006577684221171685518531273, 22.335333892828165735875678951393, 23.91557932122032153280770622949, 25.68584763147432107595705059387, 26.8497754041585174163435036840, 28.076983326652216398797951535798, 29.69483416569308317921767748891, 30.46878692734011666661714303969, 31.832162750035468301585919084421, 32.90204232055572251251634565273, 33.810627037120124508961472958701, 36.3887787775696410616030558024, 37.12576164201851730220604498195

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