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Properties

Label 1-4235-4235.2229-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.989 + 0.143i$
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  + (0.879 + 0.475i)2-s + (−0.104 + 0.994i)3-s + (0.548 + 0.836i)4-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (−0.978 − 0.207i)9-s + (−0.888 + 0.458i)12-s + (0.254 − 0.967i)13-s + (−0.398 + 0.917i)16-s + (−0.345 − 0.938i)17-s + (−0.761 − 0.647i)18-s + (0.272 + 0.962i)19-s + (0.995 + 0.0950i)23-s + (−0.999 − 0.0190i)24-s + (0.683 − 0.730i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.879 + 0.475i)2-s + (−0.104 + 0.994i)3-s + (0.548 + 0.836i)4-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (−0.978 − 0.207i)9-s + (−0.888 + 0.458i)12-s + (0.254 − 0.967i)13-s + (−0.398 + 0.917i)16-s + (−0.345 − 0.938i)17-s + (−0.761 − 0.647i)18-s + (0.272 + 0.962i)19-s + (0.995 + 0.0950i)23-s + (−0.999 − 0.0190i)24-s + (0.683 − 0.730i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.989 + 0.143i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1854374049 + 2.568628889i\)
\(L(\frac12)\) \(\approx\) \(0.1854374049 + 2.568628889i\)
\(L(1)\) \(\approx\) \(1.186814628 + 1.121748166i\)
\(L(1)\) \(\approx\) \(1.186814628 + 1.121748166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.879 + 0.475i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.254 - 0.967i)T \)
17 \( 1 + (-0.345 - 0.938i)T \)
19 \( 1 + (0.272 + 0.962i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.466 + 0.884i)T \)
31 \( 1 + (-0.161 + 0.986i)T \)
37 \( 1 + (0.625 + 0.780i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.761 + 0.647i)T \)
53 \( 1 + (0.398 + 0.917i)T \)
59 \( 1 + (0.991 + 0.132i)T \)
61 \( 1 + (-0.851 - 0.524i)T \)
67 \( 1 + (-0.235 - 0.971i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (-0.00951 - 0.999i)T \)
79 \( 1 + (-0.797 + 0.603i)T \)
83 \( 1 + (0.736 + 0.676i)T \)
89 \( 1 + (-0.928 - 0.371i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−18.193245021656869340420437040, −17.46158318606152410189224654792, −16.69223141168930483015445323207, −15.970011042407108020879017045994, −15.00405847843934694868442597015, −14.60641060062752230443758334163, −13.6916439988658746209999267895, −13.26050077206322817904536477071, −12.76933061561115026069709799964, −11.91832095788190139653078785425, −11.32014346254669910763425580063, −10.96776758102017475230663302794, −9.881648514906802596824737273487, −9.08337654080168249284748646475, −8.30944615856642743314477700207, −7.28371445718999106207717927160, −6.75556718792662391706164187133, −6.1214155904624370194062586102, −5.42548974161724729808892496611, −4.527032918273334422488171596478, −3.83371609754289825980929107056, −2.80730428373872161419867228980, −2.20346844386377637244318817897, −1.45875914793285048482472587792, −0.52656710454763999020679638149, 1.19625713823151062133822390506, 2.6313280613565567505368764531, 3.170186787305326879341224096897, 3.76938918730030111996883255792, 4.8773279195673651508802068473, 5.03790200801418738119124927033, 5.93147700090811971234261351455, 6.61831225370059015896265286508, 7.52684738951995224411840750803, 8.277255899015694763389879081996, 8.94220930008294394656033861351, 9.80632240188527598148790521204, 10.65079604396553955773567041205, 11.15393066384095437601953177833, 11.946290438164779986335060643394, 12.59305412511526618832986633989, 13.41008995896782380402001612847, 14.09593513760356667270009337740, 14.719438672530043619934618653204, 15.31313409832450518095494188254, 15.92014275047132643603869021170, 16.448512581433390651424486237312, 17.05871909591703711442129311141, 17.84880048559653904607225179028, 18.449199155784307938246351955202

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