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Properties

Label 1-61-61.36-r0-0-0
Degree $1$
Conductor $61$
Sign $-0.990 + 0.135i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.990 + 0.135i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ -0.990 + 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04055904576 + 0.5981137457i\)
\(L(\frac12)\) \(\approx\) \(0.04055904576 + 0.5981137457i\)
\(L(1)\) \(\approx\) \(0.4402258248 + 0.5873308340i\)
\(L(1)\) \(\approx\) \(0.4402258248 + 0.5873308340i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.978 + 0.207i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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