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Properties

Label 1-4235-4235.1502-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.942 + 0.335i$
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  + (0.924 − 0.380i)2-s + (−0.743 − 0.669i)3-s + (0.710 − 0.703i)4-s + (−0.941 − 0.336i)6-s + (0.389 − 0.921i)8-s + (0.104 + 0.994i)9-s + (−0.998 + 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (0.475 + 0.879i)18-s + (−0.0665 + 0.997i)19-s + (0.814 − 0.580i)23-s + (−0.905 + 0.424i)24-s + (−0.999 + 0.0190i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.924 − 0.380i)2-s + (−0.743 − 0.669i)3-s + (0.710 − 0.703i)4-s + (−0.941 − 0.336i)6-s + (0.389 − 0.921i)8-s + (0.104 + 0.994i)9-s + (−0.998 + 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (0.475 + 0.879i)18-s + (−0.0665 + 0.997i)19-s + (0.814 − 0.580i)23-s + (−0.905 + 0.424i)24-s + (−0.999 + 0.0190i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.942 + 0.335i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.942 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489298942 + 0.2573403407i\)
\(L(\frac12)\) \(\approx\) \(1.489298942 + 0.2573403407i\)
\(L(1)\) \(\approx\) \(1.192181432 - 0.4216855578i\)
\(L(1)\) \(\approx\) \(1.192181432 - 0.4216855578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.924 - 0.380i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (-0.931 - 0.362i)T \)
17 \( 1 + (-0.244 + 0.969i)T \)
19 \( 1 + (-0.0665 + 0.997i)T \)
23 \( 1 + (0.814 - 0.580i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (0.548 + 0.836i)T \)
37 \( 1 + (-0.986 + 0.161i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.475 + 0.879i)T \)
53 \( 1 + (0.999 - 0.00951i)T \)
59 \( 1 + (-0.997 - 0.0760i)T \)
61 \( 1 + (0.991 - 0.132i)T \)
67 \( 1 + (-0.690 + 0.723i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.976 - 0.217i)T \)
79 \( 1 + (-0.683 + 0.730i)T \)
83 \( 1 + (0.980 + 0.198i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.996 + 0.0855i)T \)
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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

−18.01433133326423235082253985732, −17.39243921389926674832385156513, −16.85392427601844589725826539429, −16.265407866117432240443766653410, −15.52505464812769011200672620907, −15.06639862757850019184841690758, −14.44347408498142101122943289277, −13.504198137736440280984842324307, −13.017060440224985543608059335436, −12.01446367775073871696870077390, −11.665397905623975086680746363, −11.04276674518435774018491069068, −10.24124719430566382904462054722, −9.33347410495047355555442418374, −8.80915814884811747706301113375, −7.44610542123351186219235543816, −7.113951597670871892778332208157, −6.31990706462951825580513278768, −5.47059413919862573675832750796, −4.9177845488646701690304159549, −4.43006626950564331763681116488, −3.50107008839844995906470168815, −2.79357984263939447435963122963, −1.83109755745581505887712525641, −0.33763653884912211828043266215, 1.030405366012902295510509439190, 1.7807634332009691005381467379, 2.52848007205287146831231787063, 3.43019867304880761388793060232, 4.37829276028442789035429430870, 5.05437674566627118040691090944, 5.70142429863264721941753557040, 6.37648801684279917250295881835, 7.042060958355940559939615122517, 7.72885967662500839097216251708, 8.61230708991207903139451929741, 9.84601394493148926546850815216, 10.44552093840476212570007735776, 10.94973273663531856208897442984, 11.87344461634433548144292438992, 12.244706228340560797494458738635, 12.97619968725655426637536179962, 13.32940901566868243040916886574, 14.41098656685690889349366984906, 14.7554241435362945930247340148, 15.64076610189898449649435166832, 16.42157118625014404066392005679, 17.04445258830101255970373000907, 17.65142466695505161795388511799, 18.65120798105952452744489803262

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