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Properties

Label 1-4235-4235.2657-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.627 + 0.778i$
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  + (0.0380 − 0.999i)2-s + (−0.743 − 0.669i)3-s + (−0.997 − 0.0760i)4-s + (−0.696 + 0.717i)6-s + (−0.113 + 0.993i)8-s + (0.104 + 0.994i)9-s + (0.690 + 0.723i)12-s + (0.336 + 0.941i)13-s + (0.988 + 0.151i)16-s + (−0.730 + 0.683i)17-s + (0.997 − 0.0665i)18-s + (0.483 + 0.875i)19-s + (0.458 + 0.888i)23-s + (0.749 − 0.662i)24-s + (0.953 − 0.299i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.0380 − 0.999i)2-s + (−0.743 − 0.669i)3-s + (−0.997 − 0.0760i)4-s + (−0.696 + 0.717i)6-s + (−0.113 + 0.993i)8-s + (0.104 + 0.994i)9-s + (0.690 + 0.723i)12-s + (0.336 + 0.941i)13-s + (0.988 + 0.151i)16-s + (−0.730 + 0.683i)17-s + (0.997 − 0.0665i)18-s + (0.483 + 0.875i)19-s + (0.458 + 0.888i)23-s + (0.749 − 0.662i)24-s + (0.953 − 0.299i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5278064260 + 0.2525664954i\)
\(L(\frac12)\) \(\approx\) \(0.5278064260 + 0.2525664954i\)
\(L(1)\) \(\approx\) \(0.6357356940 - 0.2941647852i\)
\(L(1)\) \(\approx\) \(0.6357356940 - 0.2941647852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0380 - 0.999i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.336 + 0.941i)T \)
17 \( 1 + (-0.730 + 0.683i)T \)
19 \( 1 + (0.483 + 0.875i)T \)
23 \( 1 + (0.458 + 0.888i)T \)
29 \( 1 + (-0.921 + 0.389i)T \)
31 \( 1 + (-0.991 - 0.132i)T \)
37 \( 1 + (0.524 - 0.851i)T \)
41 \( 1 + (0.985 - 0.170i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (-0.997 - 0.0665i)T \)
53 \( 1 + (-0.151 - 0.988i)T \)
59 \( 1 + (-0.345 - 0.938i)T \)
61 \( 1 + (0.532 + 0.846i)T \)
67 \( 1 + (0.371 + 0.928i)T \)
71 \( 1 + (0.0855 + 0.996i)T \)
73 \( 1 + (0.353 - 0.935i)T \)
79 \( 1 + (0.861 - 0.508i)T \)
83 \( 1 + (0.0570 - 0.998i)T \)
89 \( 1 + (0.327 + 0.945i)T \)
97 \( 1 + (-0.980 + 0.198i)T \)
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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

−18.18472516441504807761489718799, −17.47026287602215146280646886665, −16.81292069222940766142241146447, −16.26592240412066469475441413970, −15.5040935191960421839912986040, −15.24106543415411734804271446665, −14.42841665201193159748343428200, −13.58494929257052290023536378445, −12.89075557275588465560550340501, −12.315685558299376126879329305994, −11.15576398809057582180289853819, −10.875632303248730653057980809054, −9.792230138270182563114523119611, −9.31535229279945404731953497805, −8.65090901910533351508451960305, −7.70996032437656446447318033038, −7.00120899743430995875406025368, −6.29763277443513643566257064209, −5.61714622320189071277809661139, −4.96668369417819138502429048529, −4.39040150299190908797931642322, −3.536793419056407562884339875014, −2.69358324802717912105680663340, −1.03239943910471603011775568211, −0.229736215467775363176403836667, 1.0786379162040827596722776101, 1.757146642235334071314182118884, 2.34191810883211723815801828595, 3.59456401659842801126042274488, 4.11618943082320944391252565528, 5.11093531461344462980700202481, 5.70766327758964982734676573877, 6.45817112237936480644186786625, 7.41554551696552241940590495602, 8.041703111197808618342694554596, 9.03048871603075508020596363772, 9.533222431650439590595037207737, 10.52424900192476378486906294109, 11.18004412495081137391206489407, 11.46134519007283667645321020104, 12.3303187404214882887208867444, 12.962452793608151462396219886578, 13.35086022786205170009941730400, 14.26901828039941499429653047531, 14.76189863556673608783932720740, 16.07805445691844602799815511802, 16.521476423060890094380055591864, 17.46316722934934884673676291853, 17.82740178340577200279192447529, 18.57632862180872565115418415611

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