↑

Properties

Label 1-4235-4235.2274-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.165 + 0.986i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.564 + 0.825i)2-s + (−0.809 + 0.587i)3-s + (−0.362 − 0.931i)4-s + (−0.0285 − 0.999i)6-s + (0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 + 0.540i)12-s + (−0.774 + 0.633i)13-s + (−0.736 + 0.676i)16-s + (−0.897 + 0.441i)17-s + (0.610 + 0.791i)18-s + (−0.466 − 0.884i)19-s + (−0.415 − 0.909i)23-s + (−0.921 + 0.389i)24-s + (−0.0855 − 0.996i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.564 + 0.825i)2-s + (−0.809 + 0.587i)3-s + (−0.362 − 0.931i)4-s + (−0.0285 − 0.999i)6-s + (0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 + 0.540i)12-s + (−0.774 + 0.633i)13-s + (−0.736 + 0.676i)16-s + (−0.897 + 0.441i)17-s + (0.610 + 0.791i)18-s + (−0.466 − 0.884i)19-s + (−0.415 − 0.909i)23-s + (−0.921 + 0.389i)24-s + (−0.0855 − 0.996i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3980256981 + 0.3368304479i\)
\(L(\frac12)\) \(\approx\) \(0.3980256981 + 0.3368304479i\)
\(L(1)\) \(\approx\) \(0.4583774911 + 0.2392690065i\)
\(L(1)\) \(\approx\) \(0.4583774911 + 0.2392690065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.564 + 0.825i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.774 + 0.633i)T \)
17 \( 1 + (-0.897 + 0.441i)T \)
19 \( 1 + (-0.466 - 0.884i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
31 \( 1 + (0.254 + 0.967i)T \)
37 \( 1 + (0.998 + 0.0570i)T \)
41 \( 1 + (0.941 - 0.336i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.610 - 0.791i)T \)
53 \( 1 + (0.736 + 0.676i)T \)
59 \( 1 + (-0.941 - 0.336i)T \)
61 \( 1 + (-0.564 - 0.825i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (-0.985 + 0.170i)T \)
73 \( 1 + (-0.198 + 0.980i)T \)
79 \( 1 + (-0.516 - 0.856i)T \)
83 \( 1 + (-0.993 - 0.113i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.921 + 0.389i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.24944050644426016120105592665, −17.646990393833736295985948846828, −17.07997301847766403338627538851, −16.54058675064094925268107552779, −15.73864001563146804055752286345, −14.85053149973324253736092450947, −13.79765226229391504633451699764, −13.17133038329526181823571539325, −12.69017044515437262476609711513, −11.93528819588424611117508500712, −11.39194623960471948980960192979, −10.84133878729160105477792470714, −9.95528276183686596710917567059, −9.56249631724645371947941264099, −8.48526812301627000974657753341, −7.64080583221508167836053445672, −7.44782059080881141011490332227, −6.27676152344586991426011689246, −5.64450244755490596480667431028, −4.604747150612205958322294327529, −4.087247389763282102270115777729, −2.85566021395327679065662625139, −2.21758736355763029870295429664, −1.43017828927547542571970043559, −0.42758240539931466432491792903, 0.463060187949483909843884308920, 1.62966209876361124480305471357, 2.59488804787342218095236283077, 4.057587022520813503439073579, 4.501119798867286054354982586755, 5.223306610494266634972190634038, 5.97050645440612498529948021042, 6.80524429011808732490094643986, 7.03126485246582057378574869621, 8.21294152446328435817927822807, 9.01522225753395801711693364456, 9.37895159941233435017589761206, 10.36801010002759054780073733946, 10.73104438992679390775854560426, 11.484232020285193260552145088141, 12.35268350940515881039671434917, 13.09642903205104660247057047496, 14.03783157706106050761346791029, 14.76345325484974751118042025875, 15.25173970001867096753714570258, 15.97155584921269898405907909320, 16.51839979154080344987318207130, 17.21649551925176808012202689644, 17.53408127643515222522743681608, 18.39792853044453360863041762089

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