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Properties

Label 1-4235-4235.3358-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

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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: $\chi_{4235} (3358, \cdot )$
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 \( 1 + (-0.825 - 0.564i)T \)
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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

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

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