Properties

Label 1-421-421.414-r0-0-0
Degree $1$
Conductor $421$
Sign $-0.352 + 0.935i$
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  + (−0.691 + 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.0448 + 0.998i)5-s + (−0.900 + 0.433i)6-s + (0.936 − 0.351i)7-s + (0.753 + 0.657i)8-s + (0.753 + 0.657i)9-s + (−0.691 − 0.722i)10-s + (−0.550 − 0.834i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.393 + 0.919i)14-s + (−0.393 + 0.919i)15-s + (−0.995 + 0.0896i)16-s + (0.134 + 0.990i)17-s + ⋯
L(s)  = 1  + (−0.691 + 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.0448 + 0.998i)5-s + (−0.900 + 0.433i)6-s + (0.936 − 0.351i)7-s + (0.753 + 0.657i)8-s + (0.753 + 0.657i)9-s + (−0.691 − 0.722i)10-s + (−0.550 − 0.834i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.393 + 0.919i)14-s + (−0.393 + 0.919i)15-s + (−0.995 + 0.0896i)16-s + (0.134 + 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $-0.352 + 0.935i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (414, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ -0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7621365857 + 1.100950986i\)
\(L(\frac12)\) \(\approx\) \(0.7621365857 + 1.100950986i\)
\(L(1)\) \(\approx\) \(0.9026643778 + 0.6113606373i\)
\(L(1)\) \(\approx\) \(0.9026643778 + 0.6113606373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (-0.691 + 0.722i)T \)
3 \( 1 + (0.936 + 0.351i)T \)
5 \( 1 + (-0.0448 + 0.998i)T \)
7 \( 1 + (0.936 - 0.351i)T \)
11 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.134 + 0.990i)T \)
19 \( 1 + (0.983 + 0.178i)T \)
23 \( 1 + (-0.963 + 0.266i)T \)
29 \( 1 + T \)
31 \( 1 + (0.858 + 0.512i)T \)
37 \( 1 + (-0.550 - 0.834i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (-0.995 + 0.0896i)T \)
47 \( 1 + (-0.0448 + 0.998i)T \)
53 \( 1 + (0.936 - 0.351i)T \)
59 \( 1 + (0.473 + 0.880i)T \)
61 \( 1 + (-0.995 + 0.0896i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.995 - 0.0896i)T \)
83 \( 1 + (0.473 + 0.880i)T \)
89 \( 1 + (0.858 - 0.512i)T \)
97 \( 1 + (0.983 - 0.178i)T \)
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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

−24.50208916652868172459777152, −23.09850612071787744301224032187, −21.79667427267005917003959064489, −20.966459258081926431003477401087, −20.23719384597098516791714569462, −20.02578749415688812375685974916, −18.747827603150890749766789505049, −17.95159732993033741294274068332, −17.41710015197352376241100629766, −16.06155234058122695451843037168, −15.30083652940110517412291425436, −14.01607996732629630367725796237, −13.2144348234369767011712251865, −12.11395089746650031668556715534, −11.904423172998512187837175849023, −10.133570035994192415956714540961, −9.581843884006245981858245723842, −8.50998852661620657880399945031, −7.93888904802660869274619222525, −7.19726080722510751988504458615, −5.12226948225983617952925890945, −4.33438216346853580325665019074, −2.83768047009794056697618024682, −2.060322568330503523050934333814, −0.93741789024508281925255314966, 1.58855823558810948075387336449, 2.69856453156437412073791731, 4.03714175695404655740100997641, 5.18261218408168528682632004503, 6.429433211494970464802587553750, 7.64001118340431600219003616528, 7.93256168678294950914495530386, 9.03408525917059875028397303286, 10.20304115782240790868343637251, 10.58235617314280167283200157467, 11.75980141995597005432410315833, 13.707491913810382430077201267142, 14.13678493692118444981728795957, 14.80861575687650471255097322079, 15.66964285301438978535275230609, 16.49051440389109259449438743187, 17.69623039286173618041336383646, 18.32674399969523266242853710842, 19.36816108323632691141034923022, 19.720899151839131494307302917098, 21.09968587673055315735992709457, 21.70207359893845439288387430751, 22.91392713911513943338014527915, 24.066007247800219092332476462663, 24.41982210827516005909445026439

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