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Properties

Label 1-421-421.152-r0-0-0
Degree $1$
Conductor $421$
Sign $-0.0980 - 0.995i$
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  + (−0.900 − 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (−0.900 + 0.433i)6-s + (0.623 + 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + 12-s + 13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (−0.222 + 0.974i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (−0.900 + 0.433i)6-s + (0.623 + 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + 12-s + 13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (−0.222 + 0.974i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $-0.0980 - 0.995i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ -0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9031618069 - 0.9964876882i\)
\(L(\frac12)\) \(\approx\) \(0.9031618069 - 0.9964876882i\)
\(L(1)\) \(\approx\) \(0.9230297629 - 0.5456575040i\)
\(L(1)\) \(\approx\) \(0.9230297629 - 0.5456575040i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−24.85581858109902632657537689451, −23.65168964901783140042475792627, −22.77689028178678160710685969447, −21.68925771971503840985931003628, −20.744414608570306071896567793839, −20.15574814489003062806530706775, −19.25192038013545394070876193334, −18.24446730754929716403627582135, −17.37240819863221344425483989442, −16.84022756056359637996183795956, −15.441634306172642951577160678142, −15.10020913870834285020280531665, −14.056672248518439827335148320602, −13.49866120449717359767122344929, −11.3349850901066706154341217475, −10.84248310887310781840594463322, −9.9811903302749527653966167887, −9.195617520025980858341200154584, −8.309437093131580402709437442001, −7.20219841831293408034603206679, −6.46267817904524496976017263706, −5.08296915579126974913585831562, −3.975325255033145309426185264008, −2.53874374031429503990262385679, −1.56128059585803952827397726813, 1.100558796358181213054654615802, 1.841598805004692212265419581367, 2.85692823897898305640499986141, 4.21852146496322169961193501361, 5.9822706103762476309486781804, 6.6357806818305696334896663093, 8.23325764471328628006316582329, 8.65732138288098210363580317238, 9.09080077187859312000869745995, 10.552800380904124946732392916535, 11.552105678081459136884594924511, 12.45992271776279022809002750037, 13.13865562068668265775359554861, 14.156025044992167417267317369489, 15.27471087946594579666013941317, 16.39312950207049572931620157076, 17.29926598515776943198126461243, 18.07629493361916693822470269713, 18.69845599851360323996711659438, 19.64328927567114725587932183385, 20.349052281063651954383786936716, 21.33756444887238868420231633000, 21.62340383054213916909420238862, 23.39682748569753145954600272598, 24.49169330445266241958935614699

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