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Properties

Label 1-4235-4235.1193-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.847 + 0.530i$
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  + (0.983 + 0.179i)2-s + (−0.743 − 0.669i)3-s + (0.935 + 0.353i)4-s + (−0.610 − 0.791i)6-s + (0.856 + 0.516i)8-s + (0.104 + 0.994i)9-s + (−0.458 − 0.888i)12-s + (0.0570 + 0.998i)13-s + (0.749 + 0.662i)16-s + (0.0380 + 0.999i)17-s + (−0.0760 + 0.997i)18-s + (0.345 − 0.938i)19-s + (0.0950 − 0.995i)23-s + (−0.290 − 0.956i)24-s + (−0.123 + 0.992i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.983 + 0.179i)2-s + (−0.743 − 0.669i)3-s + (0.935 + 0.353i)4-s + (−0.610 − 0.791i)6-s + (0.856 + 0.516i)8-s + (0.104 + 0.994i)9-s + (−0.458 − 0.888i)12-s + (0.0570 + 0.998i)13-s + (0.749 + 0.662i)16-s + (0.0380 + 0.999i)17-s + (−0.0760 + 0.997i)18-s + (0.345 − 0.938i)19-s + (0.0950 − 0.995i)23-s + (−0.290 − 0.956i)24-s + (−0.123 + 0.992i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.847 + 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.600283003 + 0.7462117013i\)
\(L(\frac12)\) \(\approx\) \(2.600283003 + 0.7462117013i\)
\(L(1)\) \(\approx\) \(1.609097745 + 0.09826142576i\)
\(L(1)\) \(\approx\) \(1.609097745 + 0.09826142576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.983 + 0.179i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.0570 + 0.998i)T \)
17 \( 1 + (0.0380 + 0.999i)T \)
19 \( 1 + (0.345 - 0.938i)T \)
23 \( 1 + (0.0950 - 0.995i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (-0.988 + 0.151i)T \)
37 \( 1 + (0.263 - 0.964i)T \)
41 \( 1 + (0.0285 + 0.999i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (0.0760 + 0.997i)T \)
53 \( 1 + (0.662 + 0.749i)T \)
59 \( 1 + (0.879 - 0.475i)T \)
61 \( 1 + (0.761 - 0.647i)T \)
67 \( 1 + (0.971 - 0.235i)T \)
71 \( 1 + (0.696 - 0.717i)T \)
73 \( 1 + (-0.803 + 0.595i)T \)
79 \( 1 + (-0.820 - 0.572i)T \)
83 \( 1 + (0.491 + 0.870i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.226 + 0.974i)T \)
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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

−18.26374987469514847340955834908, −17.54783821861633949984279951289, −16.68936968316014002364883878741, −16.18072665776054416857110224162, −15.59488551180691780077846811508, −14.90912076651873249932334741709, −14.39651460525503482381016933500, −13.438107539090143774722007003624, −12.85494559762129241985284937940, −12.114371840786190594253999821072, −11.48837604790960691852972652770, −11.01279124380868380201180253030, −10.09886822342468641442717386474, −9.79011055480859345576521187923, −8.72906177167354080632183130437, −7.532481481781822009408395984227, −7.0775079407288504450125918266, −6.0334404678035511423578399917, −5.40211342224292799522121006985, −5.16674188410444158501462266022, −4.00880350280478321479817204244, −3.56220135303313890218469317032, −2.77601631161274687118267738019, −1.63872585017997580724778553636, −0.66858033664249712006462628888, 0.948318386591766093858638909307, 1.988281721536390197669167959786, 2.442593816846721695902624818001, 3.69109960947696271125960145920, 4.35461293641727029880834219872, 5.106389505715252721162303764702, 5.84196527245101928505939913748, 6.44479530708332067385308416444, 7.07049069178349228040231620589, 7.6848964903627633333651383721, 8.54243563170235788749066032405, 9.50466199825766629828375932420, 10.691792709009450703986931281756, 11.05606809996594095300202109293, 11.70140544950441205024029046240, 12.48180341108453287179158903423, 12.91617497510882870056038955811, 13.56120135958689107245284780921, 14.333532777045491327955512771060, 14.85554703410153878289863261502, 15.87302666875146615501313350366, 16.326837543630326921284474002395, 17.0415202537462595288193094224, 17.51755243584393612366458749381, 18.47627995579998668859247706001

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