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Properties

Label 1-421-421.33-r0-0-0
Degree $1$
Conductor $421$
Sign $0.611 + 0.790i$
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  + (0.623 − 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)11-s + 12-s + 13-s + (−0.900 − 0.433i)14-s + (−0.900 − 0.433i)15-s + (−0.900 + 0.433i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)11-s + 12-s + 13-s + (−0.900 − 0.433i)14-s + (−0.900 − 0.433i)15-s + (−0.900 + 0.433i)16-s + (0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.611 + 0.790i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.611 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178776751 + 0.5784233307i\)
\(L(\frac12)\) \(\approx\) \(1.178776751 + 0.5784233307i\)
\(L(1)\) \(\approx\) \(1.161971883 + 0.07893803653i\)
\(L(1)\) \(\approx\) \(1.161971883 + 0.07893803653i\)

Euler product

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

−24.18470243632010527565433223243, −23.428625367021246662800616188380, −22.64527481306671910489223654141, −21.65545182007245976414185487172, −20.81213919490120726008551505014, −19.76319185686528923306070225132, −18.52457477506760748341703358231, −18.12735531679245379998554242155, −16.85083140115236054487268099784, −16.197216656854580004953359264629, −15.55949409174060907548864755998, −14.13047550265982279528971072081, −13.50578536129894782282117667372, −12.65056875326528198749688232424, −12.01587151113914008838216367116, −11.20625421951102014456203557746, −9.07264907100478901391772897851, −8.553865150535613691679443852339, −7.71770394996690577895346501911, −6.54182661554584795869727476908, −5.65015790587800764966907615904, −5.11588501884942590004934192427, −3.54510081110692089161886301324, −2.4388134071895356937146772464, −0.66959689662646491339869467799, 1.4962555237429426535042978806, 3.20203890781427992495644942889, 3.64355225118416217546116395221, 4.59774804949768668394917190941, 5.79725224144883753034251698375, 6.707091298003350142001152991773, 8.071607068611327082897220723146, 9.76566536925175679883407763472, 10.09272205017332792018550321410, 10.93508524226245545162719497956, 11.63717862856143787395224804396, 12.779359202798157734225168216331, 13.954580900139889156905400619536, 14.53379367443614083347829831449, 15.4349274603957888985986698058, 16.200097004667974629469293207290, 17.535989914472071440755680122936, 18.321378478601792727110155264797, 19.49046965692483003900558814597, 20.18107792113569357013505183136, 20.978411832585636379658639881915, 21.73269240000214345724572301749, 22.68510814129662460868065229786, 23.206359120559577681261227211100, 23.6255503180156194243336754251

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