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Properties

Label 1-4235-4235.1028-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.808 - 0.588i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.0570 + 0.998i)2-s + (0.951 − 0.309i)3-s + (−0.993 + 0.113i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (0.809 − 0.587i)9-s + (−0.909 + 0.415i)12-s + (−0.491 + 0.870i)13-s + (0.974 − 0.226i)16-s + (−0.336 + 0.941i)17-s + (0.633 + 0.774i)18-s + (−0.0285 − 0.999i)19-s + (−0.755 − 0.654i)23-s + (−0.466 − 0.884i)24-s + (−0.897 − 0.441i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.0570 + 0.998i)2-s + (0.951 − 0.309i)3-s + (−0.993 + 0.113i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (0.809 − 0.587i)9-s + (−0.909 + 0.415i)12-s + (−0.491 + 0.870i)13-s + (0.974 − 0.226i)16-s + (−0.336 + 0.941i)17-s + (0.633 + 0.774i)18-s + (−0.0285 − 0.999i)19-s + (−0.755 − 0.654i)23-s + (−0.466 − 0.884i)24-s + (−0.897 − 0.441i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.808 - 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467337460 - 0.4772131604i\)
\(L(\frac12)\) \(\approx\) \(1.467337460 - 0.4772131604i\)
\(L(1)\) \(\approx\) \(1.164391485 + 0.3015538323i\)
\(L(1)\) \(\approx\) \(1.164391485 + 0.3015538323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0570 + 0.998i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (-0.491 + 0.870i)T \)
17 \( 1 + (-0.336 + 0.941i)T \)
19 \( 1 + (-0.0285 - 0.999i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (-0.198 - 0.980i)T \)
37 \( 1 + (-0.676 + 0.736i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.633 + 0.774i)T \)
53 \( 1 + (-0.226 + 0.974i)T \)
59 \( 1 + (-0.254 - 0.967i)T \)
61 \( 1 + (0.998 + 0.0570i)T \)
67 \( 1 + (0.540 - 0.841i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.856 - 0.516i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.996 - 0.0855i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.884 - 0.466i)T \)
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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

−18.33539846661216747185263994440, −18.20785150517622223116430941134, −17.27562711979855546001692170617, −16.260919613986780073556821142445, −15.72256260415203876387733676716, −14.6723648279483933828535111752, −14.41172370683795663552166385444, −13.62176364065019979473665580478, −13.01740974971300713926134777075, −12.343329112964026000257511597951, −11.66211867656683592003095413769, −10.73275234010429521024855826030, −10.10407808777147443597587786451, −9.69289457909532198988695997657, −8.86595541437191789715245898696, −8.22633908659906067583999769485, −7.63107052558829474733423812285, −6.648736836119232678889342921648, −5.26215077955322382256103561175, −5.03544589847158005685922453716, −3.89598538465410365212628258155, −3.41645339501467766722414761670, −2.674958160838601929416170942059, −1.95096867586389485864578172143, −1.102732850209474633529448185757, 0.37157265299985002556392437726, 1.67642970572098539093380029253, 2.45028821888565616572710582041, 3.46406459387764901270056484769, 4.265416516438172533144166198246, 4.70675633156087525356650099815, 5.893351085949736150919665336512, 6.65556723314748065895937487793, 7.02605340340463284118215844288, 8.07556687322527913247858930015, 8.31918827087200724101051540273, 9.238233915044309400063164731163, 9.66665083038025796958199712731, 10.51321105193880310484303639032, 11.67363490043209323502818656657, 12.49338879711174853238524919286, 13.042322840354718585088888223700, 13.879468401333035653546455947072, 14.12468390199811242559503739482, 15.06613263615622880899937893378, 15.384350081419399589652547012999, 16.13768960896306424621752596741, 17.019857840742640682370765351180, 17.50248690824984741733878814394, 18.29444795732765777281156622207

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