Properties

Label 1-4235-4235.367-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.101 - 0.994i$
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  + (−0.803 − 0.595i)2-s + (−0.994 − 0.104i)3-s + (0.290 + 0.956i)4-s + (0.736 + 0.676i)6-s + (0.336 − 0.941i)8-s + (0.978 + 0.207i)9-s + (−0.189 − 0.981i)12-s + (0.856 + 0.516i)13-s + (−0.830 + 0.556i)16-s + (−0.353 + 0.935i)17-s + (−0.662 − 0.749i)18-s + (0.548 − 0.836i)19-s + (−0.618 + 0.786i)23-s + (−0.432 + 0.901i)24-s + (−0.380 − 0.924i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)2-s + (−0.994 − 0.104i)3-s + (0.290 + 0.956i)4-s + (0.736 + 0.676i)6-s + (0.336 − 0.941i)8-s + (0.978 + 0.207i)9-s + (−0.189 − 0.981i)12-s + (0.856 + 0.516i)13-s + (−0.830 + 0.556i)16-s + (−0.353 + 0.935i)17-s + (−0.662 − 0.749i)18-s + (0.548 − 0.836i)19-s + (−0.618 + 0.786i)23-s + (−0.432 + 0.901i)24-s + (−0.380 − 0.924i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.101 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4253540172 - 0.4711345317i\)
\(L(\frac12)\) \(\approx\) \(0.4253540172 - 0.4711345317i\)
\(L(1)\) \(\approx\) \(0.5363685279 - 0.1506680235i\)
\(L(1)\) \(\approx\) \(0.5363685279 - 0.1506680235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.803 - 0.595i)T \)
3 \( 1 + (-0.994 - 0.104i)T \)
13 \( 1 + (0.856 + 0.516i)T \)
17 \( 1 + (-0.353 + 0.935i)T \)
19 \( 1 + (0.548 - 0.836i)T \)
23 \( 1 + (-0.618 + 0.786i)T \)
29 \( 1 + (0.362 - 0.931i)T \)
31 \( 1 + (-0.123 - 0.992i)T \)
37 \( 1 + (-0.572 + 0.820i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (0.662 - 0.749i)T \)
53 \( 1 + (-0.556 + 0.830i)T \)
59 \( 1 + (0.00951 + 0.999i)T \)
61 \( 1 + (0.398 - 0.917i)T \)
67 \( 1 + (-0.0950 - 0.995i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (-0.846 - 0.532i)T \)
79 \( 1 + (-0.879 - 0.475i)T \)
83 \( 1 + (0.170 - 0.985i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.825 + 0.564i)T \)
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

−18.30661152822903880599614090993, −17.87225508660193449324605919823, −17.32055623426772475560377821261, −16.38630572066951067333189854217, −15.9196902524263591802396813980, −15.75041527515673720357840257706, −14.48026226829244893678386127674, −14.1022884018287177727696577012, −13.025598147088155014396823079808, −12.28714727662582290302903993407, −11.52632648781058385792902006492, −10.83637220075401924960764701654, −10.356999026363914571065078855465, −9.66559533824525742768078452122, −8.848768409692404858116041459475, −8.17392808892919426724272941582, −7.23093232531166026177392785564, −6.80766590114155278239138116827, −5.88302998695196819559169588175, −5.49348266780958122563188531049, −4.68088165144058351365287304018, −3.7589329400878071893162702469, −2.57672086127228642552193746994, −1.44823717151016359170499018930, −0.81630399173544146958317771113, 0.37271077089985208392853964863, 1.37808797182949212230108261149, 1.96017587530349836815310359983, 3.05878160397302216137208049255, 4.06175673797508696313880714868, 4.4880988963112225117660957150, 5.78622429324628809518032465250, 6.27585918760290157226360828084, 7.12411751519089446032558629393, 7.77626406929179868522687503620, 8.57769821942826725742562787602, 9.41434071133114573451220974559, 9.963394365954008987994414110670, 10.8673548650187554718205026100, 11.18602414127437315671773360468, 11.90243223582996329852476248593, 12.44444258420842860368997736986, 13.4338207352476439968661134215, 13.62504923271143419187166535043, 15.20294213701145166789577552840, 15.70726387695618667805648470354, 16.35721388861994927225445035872, 17.033135160912492799747240191645, 17.6567739184979857176503682825, 18.019544298692563477603635503089

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