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Properties

Label 1-281-281.101-r0-0-0
Degree $1$
Conductor $281$
Sign $0.940 + 0.340i$
Analytic cond. $1.30495$
Root an. cond. $1.30495$
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  + (0.134 + 0.990i)2-s + (−0.691 − 0.722i)3-s + (−0.963 + 0.266i)4-s + (0.858 − 0.512i)5-s + (0.623 − 0.781i)6-s + (−0.809 + 0.587i)7-s + (−0.393 − 0.919i)8-s + (−0.0448 + 0.998i)9-s + (0.623 + 0.781i)10-s + (0.983 + 0.178i)11-s + (0.858 + 0.512i)12-s + (−0.550 − 0.834i)13-s + (−0.691 − 0.722i)14-s + (−0.963 − 0.266i)15-s + (0.858 − 0.512i)16-s + (−0.550 + 0.834i)17-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (−0.691 − 0.722i)3-s + (−0.963 + 0.266i)4-s + (0.858 − 0.512i)5-s + (0.623 − 0.781i)6-s + (−0.809 + 0.587i)7-s + (−0.393 − 0.919i)8-s + (−0.0448 + 0.998i)9-s + (0.623 + 0.781i)10-s + (0.983 + 0.178i)11-s + (0.858 + 0.512i)12-s + (−0.550 − 0.834i)13-s + (−0.691 − 0.722i)14-s + (−0.963 − 0.266i)15-s + (0.858 − 0.512i)16-s + (−0.550 + 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 281 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(281\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(1.30495\)
Root analytic conductor: \(1.30495\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{281} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 281,\ (0:\ ),\ 0.940 + 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9983881788 + 0.1749467995i\)
\(L(\frac12)\) \(\approx\) \(0.9983881788 + 0.1749467995i\)
\(L(1)\) \(\approx\) \(0.9051902536 + 0.1948539888i\)
\(L(1)\) \(\approx\) \(0.9051902536 + 0.1948539888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad281 \( 1 \)
good2 \( 1 + (0.134 + 0.990i)T \)
3 \( 1 + (-0.691 - 0.722i)T \)
5 \( 1 + (0.858 - 0.512i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.983 + 0.178i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (-0.550 + 0.834i)T \)
19 \( 1 + (0.858 - 0.512i)T \)
23 \( 1 + (0.473 - 0.880i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.983 - 0.178i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.983 - 0.178i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.995 - 0.0896i)T \)
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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

−25.89263987056970404707469219785, −24.5504022478329188430353074637, −23.18131513469600489576602965087, −22.58940197442253343242625507985, −21.94258487831948794959448344990, −21.23586109251676144123717404467, −20.22746584811456204741302528011, −19.29637509324862528637173022305, −18.26831667331183652378932915963, −17.27710691363016636857200671508, −16.70992960206644650059786504826, −15.31201640732756894563333148994, −14.04677184069988972274685422968, −13.60195010529676650752401789024, −12.10819815663324570114434404051, −11.47176200109316323412639274037, −10.38143625150847499357683722600, −9.64830476998960379889029248916, −9.178183957590761615842328671445, −6.93248815470734242341272028489, −6.03117863394685569477294241723, −4.83868250215509281764842032146, −3.7978314694967231112547359414, −2.78303238698674911457518287600, −1.1285488468900411492491946301, 0.93923463433229279484475410694, 2.67925099664214738020544120044, 4.5501350499353045938692239458, 5.561142471421248773711626696070, 6.286558133386047690573854622122, 7.03563774668512126473572572795, 8.42023725165596947525217697598, 9.29759987420249756019988801302, 10.343952360048377526864752556768, 12.10848580203346412374932123983, 12.72065578636059071860992465873, 13.44751838049676637909268085477, 14.50096192107196085238790128975, 15.69397467189814039216426817079, 16.621584292112724616995882861767, 17.38927084793811464558611527901, 17.91991759428365210932792914903, 19.02250550157002258536428973246, 19.99011999276903436354464301132, 21.75016063248430804358558395868, 22.20556251638254020506832185166, 22.92305675507419456264068988171, 24.14181019932225452617412740673, 24.85288900753779246800195233682, 25.17587554857292080789189564783

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