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Properties

Label 1-421-421.296-r0-0-0
Degree $1$
Conductor $421$
Sign $0.996 + 0.0875i$
Analytic cond. $1.95511$
Root an. cond. $1.95511$
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  + (−0.995 + 0.0896i)2-s + (0.134 + 0.990i)3-s + (0.983 − 0.178i)4-s + (0.983 + 0.178i)5-s + (−0.222 − 0.974i)6-s + (0.134 − 0.990i)7-s + (−0.963 + 0.266i)8-s + (−0.963 + 0.266i)9-s + (−0.995 − 0.0896i)10-s + (−0.691 − 0.722i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + (−0.0448 + 0.998i)14-s + (−0.0448 + 0.998i)15-s + (0.936 − 0.351i)16-s + (0.858 − 0.512i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.134 + 0.990i)3-s + (0.983 − 0.178i)4-s + (0.983 + 0.178i)5-s + (−0.222 − 0.974i)6-s + (0.134 − 0.990i)7-s + (−0.963 + 0.266i)8-s + (−0.963 + 0.266i)9-s + (−0.995 − 0.0896i)10-s + (−0.691 − 0.722i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + (−0.0448 + 0.998i)14-s + (−0.0448 + 0.998i)15-s + (0.936 − 0.351i)16-s + (0.858 − 0.512i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.996 + 0.0875i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.996 + 0.0875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9944104579 + 0.04363054017i\)
\(L(\frac12)\) \(\approx\) \(0.9944104579 + 0.04363054017i\)
\(L(1)\) \(\approx\) \(0.8371033008 + 0.1193250912i\)
\(L(1)\) \(\approx\) \(0.8371033008 + 0.1193250912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (0.134 + 0.990i)T \)
5 \( 1 + (0.983 + 0.178i)T \)
7 \( 1 + (0.134 - 0.990i)T \)
11 \( 1 + (-0.691 - 0.722i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.858 - 0.512i)T \)
19 \( 1 + (0.753 + 0.657i)T \)
23 \( 1 + (0.473 - 0.880i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.550 + 0.834i)T \)
37 \( 1 + (-0.691 - 0.722i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (0.936 - 0.351i)T \)
47 \( 1 + (0.983 + 0.178i)T \)
53 \( 1 + (0.134 - 0.990i)T \)
59 \( 1 + (-0.393 - 0.919i)T \)
61 \( 1 + (0.936 - 0.351i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.936 + 0.351i)T \)
83 \( 1 + (-0.393 - 0.919i)T \)
89 \( 1 + (-0.550 - 0.834i)T \)
97 \( 1 + (0.753 - 0.657i)T \)
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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

−24.36039780206183530408591530775, −23.838182963206101879119805835703, −22.33141268771601645377341940880, −21.336427423639245274148501372918, −20.688956925423882799388104640671, −19.57842765985191924299809135049, −18.88903396896082992583532438658, −18.05477706124987119832996817912, −17.56190023244094029727279558089, −16.757736111731776202617489242010, −15.47416171385048370682145023015, −14.60704534671368834246510497654, −13.48509799901609540690673648723, −12.37466743816787243471613289305, −11.99052024126156062573199836784, −10.67684872312984231035233200135, −9.507545946418258801772339143455, −9.03721071709385677086852144995, −7.8448209753532889861740477409, −7.122981623224959669013915838845, −6.008056854169724990420808840270, −5.23824933367130395792467954243, −2.82857286389163316997152068299, −2.2269848478136554455960339982, −1.26900590156050857262620339753, 0.86833110970309725214570432816, 2.54055402632361447094735357688, 3.32593821254664486086175751536, 5.06201064303020554507837648971, 5.778952353521728258754835621004, 7.12058650790454136456412008351, 8.05359970531051767211789927611, 9.08211911934216693088721233757, 10.16771704356674136105406320286, 10.2863934629498032808675139427, 11.23903610425546579046038151541, 12.63410071082162106731308819308, 14.19744084571545149790578947832, 14.3877325091597191980782129063, 15.857345014169041041347364536929, 16.44793521928714163784435751177, 17.22891848020164535637714659015, 17.93508142215875959248115861704, 18.997818772384410002551332922605, 20.035366782190441305461370166, 20.80238896225752559816622263074, 21.247226712960039057761110804766, 22.36620542912107862294714321761, 23.35489709577378152944618611672, 24.61245769685756501146698569497

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