Properties

Label 1-4235-4235.193-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.523 + 0.851i$
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  + (−0.244 − 0.969i)2-s + (0.743 + 0.669i)3-s + (−0.879 + 0.475i)4-s + (0.466 − 0.884i)6-s + (0.676 + 0.736i)8-s + (0.104 + 0.994i)9-s + (−0.971 − 0.235i)12-s + (−0.791 − 0.610i)13-s + (0.548 − 0.836i)16-s + (0.572 + 0.820i)17-s + (0.938 − 0.345i)18-s + (0.797 − 0.603i)19-s + (−0.998 + 0.0475i)23-s + (0.00951 + 0.999i)24-s + (−0.398 + 0.917i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.244 − 0.969i)2-s + (0.743 + 0.669i)3-s + (−0.879 + 0.475i)4-s + (0.466 − 0.884i)6-s + (0.676 + 0.736i)8-s + (0.104 + 0.994i)9-s + (−0.971 − 0.235i)12-s + (−0.791 − 0.610i)13-s + (0.548 − 0.836i)16-s + (0.572 + 0.820i)17-s + (0.938 − 0.345i)18-s + (0.797 − 0.603i)19-s + (−0.998 + 0.0475i)23-s + (0.00951 + 0.999i)24-s + (−0.398 + 0.917i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.523 + 0.851i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.523 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3178167859 + 0.5685909428i\)
\(L(\frac12)\) \(\approx\) \(0.3178167859 + 0.5685909428i\)
\(L(1)\) \(\approx\) \(0.9159195972 - 0.07008808354i\)
\(L(1)\) \(\approx\) \(0.9159195972 - 0.07008808354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.244 + 0.969i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.791 + 0.610i)T \)
17 \( 1 + (-0.572 - 0.820i)T \)
19 \( 1 + (-0.797 + 0.603i)T \)
23 \( 1 + (0.998 - 0.0475i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (0.761 + 0.647i)T \)
37 \( 1 + (0.901 - 0.432i)T \)
41 \( 1 + (0.897 - 0.441i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.938 + 0.345i)T \)
53 \( 1 + (0.836 - 0.548i)T \)
59 \( 1 + (-0.0665 - 0.997i)T \)
61 \( 1 + (0.272 + 0.962i)T \)
67 \( 1 + (-0.618 - 0.786i)T \)
71 \( 1 + (-0.974 + 0.226i)T \)
73 \( 1 + (-0.703 - 0.710i)T \)
79 \( 1 + (0.948 + 0.318i)T \)
83 \( 1 + (0.931 - 0.362i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (0.491 - 0.870i)T \)
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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

−18.18578074232250564849400911067, −17.530691217904156311097601523209, −16.79730217140654785858764224230, −16.00573797484591562876784896831, −15.546430879292879171173363061863, −14.52454689214169977237369960521, −14.12817532791609148465037625350, −13.84080883928837396424425314415, −12.759878934337499528596348705110, −12.24552887708996285124017332508, −11.46617932286349581722844284653, −10.13696305436944254314443075515, −9.68124551049992955201479692336, −9.07963082336828767114239865451, −8.21864661787280371076861057055, −7.702235442067442546536113814815, −7.096318084785410491872215158287, −6.47075047980334631545285438724, −5.617924796395793021888443736930, −4.8644964676173912210345093948, −3.92336481561069456940401785571, −3.18076011319571080325314246416, −2.08265469445865464599393940704, −1.32782351769432421117962641898, −0.17527010217782251963412383452, 1.27814029945877910020794599986, 2.09847568009405586847722420265, 2.909671183753906502333259253314, 3.466366226473972727184665978907, 4.1972262804647541354390369700, 5.044015268892663272902363477254, 5.56687800014687268400074726396, 7.024500264340287645919330387177, 7.86587928351952429324152967789, 8.3022388294817256704229694775, 9.12690742854557566870235020332, 9.802562566016330483622564165935, 10.22583081626892672699226497315, 10.90440125130889183035253721759, 11.70640862218063989176469734315, 12.501243649556215684129386009903, 13.04029742663454316787228122959, 13.96344997873994514946576549037, 14.316107881748223597189444773682, 15.165029548043631320846286935009, 15.84899365577611318307466494776, 16.72498073265794684753554461229, 17.26216348154405892880152370905, 18.09465682225788949687271462795, 18.74578615565103419933606664238

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