Properties

Degree 1
Conductor 61
Sign $-0.990 - 0.135i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.104 − 0.994i)2-s + (−0.809 − 0.587i)3-s + (−0.978 − 0.207i)4-s + (0.669 − 0.743i)5-s + (−0.669 + 0.743i)6-s + (−0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.669 − 0.743i)10-s − 11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.978 + 0.207i)17-s + ⋯
L(s,χ)  = 1  + (0.104 − 0.994i)2-s + (−0.809 − 0.587i)3-s + (−0.978 − 0.207i)4-s + (0.669 − 0.743i)5-s + (−0.669 + 0.743i)6-s + (−0.913 − 0.406i)7-s + (−0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.669 − 0.743i)10-s − 11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.978 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.990 - 0.135i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.990 - 0.135i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $-0.990 - 0.135i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (39, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (0:\ ),\ -0.990 - 0.135i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.04055904576 - 0.5981137457i$
$L(\frac12,\chi)$  $\approx$  $0.04055904576 - 0.5981137457i$
$L(\chi,1)$  $\approx$  0.4402258248 - 0.5873308340i
$L(1,\chi)$  $\approx$  0.4402258248 - 0.5873308340i

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

−33.34749083213803027253146672027, −32.24365914950741359627814215193, −31.271117263090281455263262447553, −29.470260482806624456320260542324, −28.65482711677460597460299166821, −27.2504390511735440995579296405, −26.21018226251581389118592883072, −25.5348257115657166938231959525, −23.946393255521830959968936092994, −22.89442811391444328621862350410, −22.06795555211516577927135853758, −21.2365197854984838454300572616, −18.81042934466027837076190523927, −18.02768738265386049388085582719, −16.7358257578020555715658194170, −15.89696165277916591711433768731, −14.75816781907276535084683131088, −13.43857301188086483402540530607, −12.00450200259276953201074040087, −10.11001267950085041838180991739, −9.46470367442943960885238710330, −7.31011808368335195444334423274, −6.10165837281398055857887533714, −5.22135298789596137595188493200, −3.33392242923364637412918398936, 0.8316947264947309156519476176, 2.707572308275920172499698920690, 4.86707517021365494610280760475, 5.911484561631680426731997654327, 7.914849669000923563804989207509, 9.77283190796557386473378362919, 10.549089266244362717347772647271, 12.28496262109511726086290160249, 12.87939421654942554874570643396, 13.84608179195810264065834266196, 16.16039748088588429392120393404, 17.35078420273085936157285995461, 18.27933988747949895118393808649, 19.52329254760231938974746398843, 20.63201853869151102289971070368, 21.879741865066697257233553266458, 22.87440966512094921074295914028, 23.827553953639389431771437430685, 25.17991675040953542524117595239, 26.69498399438796218777716794447, 28.19493566689252941815959976782, 28.80480312926399458062768022079, 29.55718593734919531927737127479, 30.52968288450186482912987563382, 32.09261895842301987283249724401

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