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Properties

Label 1-4235-4235.1733-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.272 + 0.962i$
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  + (0.318 − 0.948i)2-s + (0.743 + 0.669i)3-s + (−0.797 − 0.603i)4-s + (0.870 − 0.491i)6-s + (−0.825 + 0.564i)8-s + (0.104 + 0.994i)9-s + (−0.189 − 0.981i)12-s + (0.226 − 0.974i)13-s + (0.272 + 0.962i)16-s + (−0.780 − 0.625i)17-s + (0.976 + 0.217i)18-s + (−0.935 + 0.353i)19-s + (0.618 − 0.786i)23-s + (−0.991 − 0.132i)24-s + (−0.851 − 0.524i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.318 − 0.948i)2-s + (0.743 + 0.669i)3-s + (−0.797 − 0.603i)4-s + (0.870 − 0.491i)6-s + (−0.825 + 0.564i)8-s + (0.104 + 0.994i)9-s + (−0.189 − 0.981i)12-s + (0.226 − 0.974i)13-s + (0.272 + 0.962i)16-s + (−0.780 − 0.625i)17-s + (0.976 + 0.217i)18-s + (−0.935 + 0.353i)19-s + (0.618 − 0.786i)23-s + (−0.991 − 0.132i)24-s + (−0.851 − 0.524i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.272 + 0.962i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2934679547 + 0.3882098352i\)
\(L(\frac12)\) \(\approx\) \(0.2934679547 + 0.3882098352i\)
\(L(1)\) \(\approx\) \(1.076426740 - 0.2961705012i\)
\(L(1)\) \(\approx\) \(1.076426740 - 0.2961705012i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.318 - 0.948i)T \)
3 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (0.226 - 0.974i)T \)
17 \( 1 + (-0.780 - 0.625i)T \)
19 \( 1 + (-0.935 + 0.353i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (-0.254 - 0.967i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (0.0190 + 0.999i)T \)
41 \( 1 + (-0.993 - 0.113i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (-0.976 + 0.217i)T \)
53 \( 1 + (0.962 + 0.272i)T \)
59 \( 1 + (0.595 + 0.803i)T \)
61 \( 1 + (-0.749 - 0.662i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (-0.997 + 0.0665i)T \)
79 \( 1 + (-0.179 + 0.983i)T \)
83 \( 1 + (0.884 + 0.466i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (-0.791 - 0.610i)T \)
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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

−18.011291917418917705964308746170, −17.56452462898043263139010638091, −16.76105655877091804959860331727, −16.12875896844387980333929301578, −15.15302033518436460929317850310, −14.89844598073441999059969180462, −14.17641223233049982733791231924, −13.32520714373803082735749928320, −13.14778919138770220291095315397, −12.31326436118175405813834115244, −11.549368723000041201109035554224, −10.62447348882413581749599387483, −9.44945765213451214474930873650, −8.895463912240700045907937225895, −8.54703406345395518927640795773, −7.55806579698540830648464245357, −7.000442068595731493720956523582, −6.48166065361895695690502722481, −5.69793040042047135365283439060, −4.73198326579618172948996280628, −3.90186809307117326553539946333, −3.39995173358661940892682489580, −2.25981572695759262408408377572, −1.55450093435504535955814678708, −0.101622812017106939771863966033, 1.24156872633776906445507050170, 2.23806556134450601965990691575, 2.81168651158301129659618202311, 3.516364366535562431759693809211, 4.31639463286393636136390558790, 4.86035360797716487485329375928, 5.65455256708120112851221684356, 6.5814199456247528662263254487, 7.71593083498404586385987897360, 8.54641106225300397082008080221, 8.90194463996697412235453211090, 9.83669947245635295884593844564, 10.32402392902387146509132865536, 10.95884600926932507912627537918, 11.588716655206000919570378728683, 12.60147343541529636931414090162, 13.21493288651927991175761661782, 13.61496728894521569928188724923, 14.63691037723990369164870575234, 14.931900419324863022217203316736, 15.625216186621306895415403903108, 16.49120075119200069550670461698, 17.29853387801996514347597564906, 18.12901474230657282962832849832, 18.75919789072841460011865221417

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