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Properties

Label 1-4235-4235.1083-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.479 + 0.877i$
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  + (−0.151 − 0.988i)2-s + (−0.207 + 0.978i)3-s + (−0.953 + 0.299i)4-s + (0.998 + 0.0570i)6-s + (0.441 + 0.897i)8-s + (−0.913 − 0.406i)9-s + (−0.0950 − 0.995i)12-s + (0.980 − 0.198i)13-s + (0.820 − 0.572i)16-s + (−0.132 − 0.991i)17-s + (−0.263 + 0.964i)18-s + (−0.432 + 0.901i)19-s + (−0.945 − 0.327i)23-s + (−0.969 + 0.244i)24-s + (−0.345 − 0.938i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.151 − 0.988i)2-s + (−0.207 + 0.978i)3-s + (−0.953 + 0.299i)4-s + (0.998 + 0.0570i)6-s + (0.441 + 0.897i)8-s + (−0.913 − 0.406i)9-s + (−0.0950 − 0.995i)12-s + (0.980 − 0.198i)13-s + (0.820 − 0.572i)16-s + (−0.132 − 0.991i)17-s + (−0.263 + 0.964i)18-s + (−0.432 + 0.901i)19-s + (−0.945 − 0.327i)23-s + (−0.969 + 0.244i)24-s + (−0.345 − 0.938i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6722037656 + 0.3986043876i\)
\(L(\frac12)\) \(\approx\) \(0.6722037656 + 0.3986043876i\)
\(L(1)\) \(\approx\) \(0.7463297329 - 0.07600937508i\)
\(L(1)\) \(\approx\) \(0.7463297329 - 0.07600937508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.151 - 0.988i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (0.980 - 0.198i)T \)
17 \( 1 + (-0.132 - 0.991i)T \)
19 \( 1 + (-0.432 + 0.901i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (0.0285 - 0.999i)T \)
31 \( 1 + (-0.861 + 0.508i)T \)
37 \( 1 + (-0.803 + 0.595i)T \)
41 \( 1 + (-0.774 - 0.633i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (0.263 + 0.964i)T \)
53 \( 1 + (-0.572 + 0.820i)T \)
59 \( 1 + (0.161 + 0.986i)T \)
61 \( 1 + (0.625 - 0.780i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (-0.992 - 0.123i)T \)
79 \( 1 + (0.532 - 0.846i)T \)
83 \( 1 + (-0.226 - 0.974i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.717 + 0.696i)T \)
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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

−18.20745945848169424259505668048, −17.55100236351149947443925160362, −17.026289212124300927520132681730, −16.27064935674883452615788754403, −15.66153535143259042818148587946, −14.83845727106818483220158554410, −14.17743435763185598820353162845, −13.56522447381168368083291202314, −12.91748709423281450020184613689, −12.46227842623490915397804914701, −11.326130603188691656418676522355, −10.81981438933249348432301712222, −9.875322780862872561829318973591, −8.85761389926364598506480653866, −8.493404662712822689259179900875, −7.79023770416678468572776612331, −6.94324847202631533268188506854, −6.5247111812721505703794295598, −5.76012162442476088604569530425, −5.2142990659035041838989624767, −4.11596715850422500962124857143, −3.43453068158432172674942069654, −2.08034995314245335523265508002, −1.40834362526628667806381783872, −0.31211810001706486234382781403, 0.81296463980129010163497368370, 1.92126562769251411561568763003, 2.77839496640050929095729539159, 3.63516592396519017702561560400, 4.0582042681894534404132657593, 4.90368226468682745385078219974, 5.63382274065571735603220810431, 6.35316775834519058137731402699, 7.65551598267287144394189007611, 8.407883151811070659995160667921, 9.00217069263284384630372363586, 9.677414094610521835120373418477, 10.39028448980749054509139256681, 10.80591036642336472808897328922, 11.5918187005419951314642373626, 12.11924383195965405985751570239, 12.8837696562103254192592775191, 13.97424113287046403719473876061, 14.0729787725312302042627365781, 15.15247994956923852348742529347, 15.919385801954745089391719062323, 16.432201111286569978980548509949, 17.29320441712510536290002086665, 17.79255622852768642772705074940, 18.61010885464025663461742450974

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