Properties

Label 1-4235-4235.1578-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.478 - 0.877i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.478 - 0.877i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.478 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8940173108 - 0.5308007769i\)
\(L(\frac12)\) \(\approx\) \(0.8940173108 - 0.5308007769i\)
\(L(1)\) \(\approx\) \(0.6369930201 - 0.4653393723i\)
\(L(1)\) \(\approx\) \(0.6369930201 - 0.4653393723i\)

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 \( 1 + (-0.980 + 0.198i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.07482494963750681301340056480, −17.57908844332431014259430337167, −17.17519676066036936495208395256, −16.15924294257581642803030607964, −16.08332168259465180344552089421, −15.26155931980909168304502975582, −14.50155777617308489756653369581, −14.039498094167056541896534900543, −13.11982702927414089731445345921, −12.313042524404230747120272993680, −11.74951696548036008095063690433, −10.89501523931568203473488273503, −10.00881434686727804214228057577, −9.545348615913206476761820338278, −8.86783884343290520051299062851, −8.01549396888901759317062078298, −7.15225063538837879818486018237, −6.55976889919232460207169762105, −5.83200578481382346884901991966, −5.18479500988928463094614983072, −4.50302062823879380531002656988, −3.86591355609419441595538632403, −3.02721238642013275831197707445, −1.54227293198407609407360187466, −0.46445420197115976508922679230, 0.82995115791818358596310567133, 1.26793422394579803527488439720, 2.53006301392453130465636136576, 2.876145258284916594970108310229, 4.08018105240055494906770871006, 4.82248506581366544357677513471, 5.54410859209195733446139617073, 6.11066004613905316761877306691, 7.17924203535083979413707882764, 7.99650761648131850064379539885, 8.40659384938799497780924701079, 9.66057462897475790575910365133, 10.18927857127345026828059456421, 10.69033128698051363114706890093, 11.65566651987660917864826657018, 12.08689123279173411277939683138, 12.566484711103561803743613647957, 13.30974534293343864424697335801, 14.03058322824852595601927295653, 14.513414390655123369392535919065, 15.67248192720888884734987233282, 16.41185087683707094837647045848, 17.17823237001024265951544957753, 17.765208459669895325720049572264, 18.3138695042585616656020319158

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