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Properties

Label 1-4235-4235.1307-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

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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

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

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: $\chi_{4235} (1307, \cdot )$
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 \( 1 + (-0.980 - 0.198i)T \)
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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

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

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