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Properties

Label 1-4235-4235.423-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.0214 - 0.999i$
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  + (−0.893 − 0.449i)2-s + (−0.743 − 0.669i)3-s + (0.595 + 0.803i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (0.104 + 0.994i)9-s + (0.0950 − 0.995i)12-s + (−0.491 + 0.870i)13-s + (−0.290 + 0.956i)16-s + (0.983 − 0.179i)17-s + (0.353 − 0.935i)18-s + (0.879 + 0.475i)19-s + (0.945 − 0.327i)23-s + (−0.532 + 0.846i)24-s + (0.830 − 0.556i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.893 − 0.449i)2-s + (−0.743 − 0.669i)3-s + (0.595 + 0.803i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (0.104 + 0.994i)9-s + (0.0950 − 0.995i)12-s + (−0.491 + 0.870i)13-s + (−0.290 + 0.956i)16-s + (0.983 − 0.179i)17-s + (0.353 − 0.935i)18-s + (0.879 + 0.475i)19-s + (0.945 − 0.327i)23-s + (−0.532 + 0.846i)24-s + (0.830 − 0.556i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.0214 - 0.999i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.0214 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5511197829 - 0.5394068315i\)
\(L(\frac12)\) \(\approx\) \(0.5511197829 - 0.5394068315i\)
\(L(1)\) \(\approx\) \(0.5583523066 - 0.2088491238i\)
\(L(1)\) \(\approx\) \(0.5583523066 - 0.2088491238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.893 - 0.449i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (-0.491 + 0.870i)T \)
17 \( 1 + (0.983 - 0.179i)T \)
19 \( 1 + (0.879 + 0.475i)T \)
23 \( 1 + (0.945 - 0.327i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (-0.749 + 0.662i)T \)
37 \( 1 + (-0.299 - 0.953i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.353 - 0.935i)T \)
53 \( 1 + (0.956 - 0.290i)T \)
59 \( 1 + (-0.710 + 0.703i)T \)
61 \( 1 + (-0.548 + 0.836i)T \)
67 \( 1 + (-0.998 - 0.0475i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (-0.875 - 0.483i)T \)
79 \( 1 + (0.969 - 0.244i)T \)
83 \( 1 + (0.996 - 0.0855i)T \)
89 \( 1 + (0.235 - 0.971i)T \)
97 \( 1 + (0.884 - 0.466i)T \)
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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

−18.242897765248114851400985275492, −17.85124649579599182652069190100, −17.06178450552926272349870413719, −16.64542066786898038680482417331, −16.012950414488558986868215729249, −15.18377511575266586134731893553, −14.94226898569438283295645354199, −14.05524086017169852224007821193, −12.99120290179542435402991997908, −12.1464616588544406471744628567, −11.52112388043582510888555471127, −10.80597965174625938824719906427, −10.249569322971104642301129703402, −9.57108386656899295910638754910, −9.11050181629232472874836750006, −8.065409719057720205967949727761, −7.47884477820222888948227250759, −6.66278736167039986291278437425, −5.963184060515713976784714879241, −5.116576666306123985330100085903, −4.89773229255804681537562777198, −3.4061070621137461372525472326, −2.876688870447557636529609290325, −1.440311772159568751459173832977, −0.76856793406098817943334030916, 0.476315677864130106243121481385, 1.36275455700218750945137288975, 2.0385956677715127581448019479, 2.92123259435552771957758712039, 3.82487649646992890850911698377, 4.877932492330661117163450515617, 5.65395930660203859917734681386, 6.529188386537635100653315499129, 7.30029647332003523331136994533, 7.550261980418687358171458550820, 8.5868901186524904895511531251, 9.22955585022992843477271703923, 10.15534520272476681438787644823, 10.549865284648569326919505699102, 11.5181032675260954158655529556, 11.95721778964138190635279100187, 12.395283693429210708035591494803, 13.273223609974081375578614599419, 13.97571215763418247769059004131, 14.85330466903363357018242184880, 15.902375977582752762602089255212, 16.55847129622586227916556866015, 16.82769855405167548033825116448, 17.6611693351129502068577866445, 18.265751208556162108000339710948

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