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Properties

Label 1-4235-4235.1543-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.985 - 0.167i$
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  + (0.917 + 0.398i)2-s + (0.406 − 0.913i)3-s + (0.683 + 0.730i)4-s + (0.736 − 0.676i)6-s + (0.336 + 0.941i)8-s + (−0.669 − 0.743i)9-s + (0.945 − 0.327i)12-s + (0.856 − 0.516i)13-s + (−0.0665 + 0.997i)16-s + (0.986 + 0.161i)17-s + (−0.318 − 0.948i)18-s + (0.449 − 0.893i)19-s + (−0.371 + 0.928i)23-s + (0.997 + 0.0760i)24-s + (0.991 − 0.132i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.917 + 0.398i)2-s + (0.406 − 0.913i)3-s + (0.683 + 0.730i)4-s + (0.736 − 0.676i)6-s + (0.336 + 0.941i)8-s + (−0.669 − 0.743i)9-s + (0.945 − 0.327i)12-s + (0.856 − 0.516i)13-s + (−0.0665 + 0.997i)16-s + (0.986 + 0.161i)17-s + (−0.318 − 0.948i)18-s + (0.449 − 0.893i)19-s + (−0.371 + 0.928i)23-s + (0.997 + 0.0760i)24-s + (0.991 − 0.132i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.177304983 - 0.3518341129i\)
\(L(\frac12)\) \(\approx\) \(4.177304983 - 0.3518341129i\)
\(L(1)\) \(\approx\) \(2.230514659 - 0.05686310305i\)
\(L(1)\) \(\approx\) \(2.230514659 - 0.05686310305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.917 + 0.398i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.856 - 0.516i)T \)
17 \( 1 + (0.986 + 0.161i)T \)
19 \( 1 + (0.449 - 0.893i)T \)
23 \( 1 + (-0.371 + 0.928i)T \)
29 \( 1 + (0.362 + 0.931i)T \)
31 \( 1 + (-0.797 - 0.603i)T \)
37 \( 1 + (-0.424 + 0.905i)T \)
41 \( 1 + (0.870 + 0.491i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.318 - 0.948i)T \)
53 \( 1 + (0.997 - 0.0665i)T \)
59 \( 1 + (0.861 + 0.508i)T \)
61 \( 1 + (0.595 - 0.803i)T \)
67 \( 1 + (-0.814 - 0.580i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (-0.0380 - 0.999i)T \)
79 \( 1 + (0.851 + 0.524i)T \)
83 \( 1 + (0.170 + 0.985i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
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

−18.69323543042317279449158016711, −17.74019333666297635994329741753, −16.56991395042595444776354612324, −16.15806624547484513939557391983, −15.78180132939892186643129608785, −14.604974605764513618986185045430, −14.48080293940944200166589962269, −13.81426082815020933001192213824, −13.042014229256014565518241081240, −12.18615075823751084217734183806, −11.62265777173774884693844156686, −10.77645294330873931681978480606, −10.33117925949624749018838296427, −9.60117274040743558792424690620, −8.89351087619764683257568831342, −7.98196789805278430612791240847, −7.22311380262548516732615946027, −6.05312745347354983085346683407, −5.71040379080622060615615833195, −4.79906387419605133478702960310, −4.03117441301678991020369500767, −3.6180840354828356898400408952, −2.765345181661030779639370542921, −2.002679289877159479983742239, −0.984344197452008404183316274055, 0.9362420254087235157133389555, 1.79074568740550196414712249331, 2.71063921637364556965141304135, 3.402830802273645836799162320647, 3.937419503161996540743832652409, 5.27482762576276987832276296820, 5.6257234994142829434525585602, 6.48548814322428115373767353457, 7.177092892679329695558451395319, 7.73136252311468241509357576931, 8.431307836284317808464055953, 9.10037336121192160109008275385, 10.204908337287413185436116440682, 11.17446827052337005968525600572, 11.71217471846756860652078175370, 12.45723249921643746926592709806, 12.99843015079472189064144351299, 13.73294918249967315250034521169, 14.002538763053631231065661501861, 15.01799672833870612576478719602, 15.34207330992353846879390229529, 16.29010017400572608410892022329, 16.88441384205669985505049813543, 17.85008741504672200911218719210, 18.10582513659535538642214010016

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