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Properties

Label 1-4235-4235.2958-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.225 + 0.974i$
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  + (−0.998 + 0.0475i)2-s + (0.866 − 0.5i)3-s + (0.995 − 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (0.814 − 0.580i)12-s + (−0.909 − 0.415i)13-s + (0.981 − 0.189i)16-s + (−0.971 + 0.235i)17-s + (−0.458 + 0.888i)18-s + (0.235 − 0.971i)19-s + (0.189 + 0.981i)23-s + (−0.786 + 0.618i)24-s + (0.928 + 0.371i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.998 + 0.0475i)2-s + (0.866 − 0.5i)3-s + (0.995 − 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (0.814 − 0.580i)12-s + (−0.909 − 0.415i)13-s + (0.981 − 0.189i)16-s + (−0.971 + 0.235i)17-s + (−0.458 + 0.888i)18-s + (0.235 − 0.971i)19-s + (0.189 + 0.981i)23-s + (−0.786 + 0.618i)24-s + (0.928 + 0.371i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2958, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.225 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5427327361 + 0.4314419532i\)
\(L(\frac12)\) \(\approx\) \(0.5427327361 + 0.4314419532i\)
\(L(1)\) \(\approx\) \(0.7714455427 - 0.08360746998i\)
\(L(1)\) \(\approx\) \(0.7714455427 - 0.08360746998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 + 0.0475i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (-0.971 + 0.235i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (0.189 + 0.981i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (-0.0950 + 0.995i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (0.458 + 0.888i)T \)
53 \( 1 + (-0.189 + 0.981i)T \)
59 \( 1 + (-0.0475 + 0.998i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (-0.458 + 0.888i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.945 + 0.327i)T \)
79 \( 1 + (-0.786 - 0.618i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−18.397674222145212230420907400204, −17.5561945370694906294454359574, −16.76241587303129327979525196979, −16.32941756005261416269886490842, −15.58357415345506349571823919503, −14.93367453756560481810540923480, −14.407734691292916565030057634726, −13.59244182119340913589954093547, −12.669883446054997753951270498668, −11.9901961189754912469161869713, −11.158809093519530308714260810841, −10.417024124945089394954139704685, −9.89606295329703234599326380651, −9.24254640657504692369117725970, −8.591328541902793081168185052615, −8.06618442181099841033685802770, −7.1507351154896721376939541464, −6.7569578036901345565017789247, −5.56783070558448604142290270454, −4.71535025667977408052155253295, −3.80581057975742184573098272521, −3.04881132914982680480279633984, −2.147465607601122942577511735137, −1.76841324419387166467371289963, −0.23658825206632041088078703114, 1.004368637222910160851582670299, 1.80324768121287727496126692486, 2.63018567067083353241557412929, 3.104146637698801554130257655911, 4.20532868878183508637502618448, 5.253773464331378861947657018520, 6.23853661256220985833962857187, 6.9842692757538629558396902679, 7.443800925353576650506851697695, 8.17872113392599758033029340073, 8.80216909204114610745433745499, 9.537902188114559723983698775626, 9.91830128786835834171955988764, 10.95363894769988394487926560335, 11.612843292716247049935511674979, 12.30646109418452206312832562990, 13.19783665544072412941821680161, 13.6202055908609720895626476498, 14.80032956082695561477831792997, 15.15634134171305983673141344462, 15.65379678831596223040802098658, 16.661359116730579713755136824009, 17.48575654717455295736955143496, 17.718489938001409137094683623839, 18.67034817618572278513864547224

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