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Properties

Label 1-421-421.252-r0-0-0
Degree $1$
Conductor $421$
Sign $-0.744 - 0.667i$
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  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)5-s + 6-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)5-s + 6-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ -0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2840494769 - 0.7426510625i\)
\(L(\frac12)\) \(\approx\) \(0.2840494769 - 0.7426510625i\)
\(L(1)\) \(\approx\) \(0.8213306099 - 0.3737623019i\)
\(L(1)\) \(\approx\) \(0.8213306099 - 0.3737623019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−24.64217459779101124389030154333, −23.690198226641560352251855206534, −23.27316929828977679158048817326, −22.209292246308120120943169142430, −21.21139743174529942427061811445, −20.02555561608664797820854572158, −19.295781937543095266940330873288, −18.33411577666608608710751527186, −17.49425693953564041127297175559, −16.828228790587153183175973964174, −15.41856976460192448405639778139, −15.05591704317426173937090556288, −14.17910721534434511722548776851, −12.90453627773476504576443267302, −12.27788517893547888041946734328, −11.92651684003605472656457054850, −9.899342817321487088098492458552, −8.61750115814486457447917160030, −8.213513354709389488995177337318, −7.36882644658099011752977922978, −6.3302578042070129515556757457, −5.32784961988794643474847521171, −4.31288686101767261050787032855, −3.05819207582857382552905611984, −1.60669446076456827592468021146, 0.39851382066120788719953588237, 2.398149863255692484184356888793, 3.32038214831365237175964007757, 4.23439653121884802039962967979, 4.77130014564898247804356618304, 6.3355887818897909693868881706, 7.75080970976098264233929819278, 8.72468297634108692614655072344, 9.795390201780535217583101546053, 10.59431169060193598080937150897, 11.3014552370505633065790383526, 11.95127619479475411218500637045, 13.487313994002356930972685520502, 14.30464519296692906080433693446, 14.698002973141280550809417539785, 15.95799290717151493616670432217, 16.80110368283448937956377624415, 17.949961670045491038899916092762, 19.26681855019601292729275028278, 19.53430161091587407113687987542, 20.47416465262242657059346032016, 21.277927544084839419940555373888, 22.07977379074684981435683893349, 22.7545825648875298036163012606, 23.61407658396710195838178659248

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