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Properties

Label 1-4600-4600.3219-r0-0-0
Degree $1$
Conductor $4600$
Sign $-0.929 - 0.368i$
Analytic cond. $21.3623$
Root an. cond. $21.3623$
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  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (3219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01111261141 + 0.05825434908i\)
\(L(\frac12)\) \(\approx\) \(0.01111261141 + 0.05825434908i\)
\(L(1)\) \(\approx\) \(0.7044119138 + 0.1509837955i\)
\(L(1)\) \(\approx\) \(0.7044119138 + 0.1509837955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 - T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.86837649356112377934077301771, −17.08932679882838418154556817478, −16.65511422348664599237251513422, −16.05764563573146794345375609319, −15.03856156744884217372784592302, −14.28557486941371881775031216781, −13.85361815650759794840478574865, −12.9636641366928620109599869271, −12.304684880233849975456059478846, −12.05013014836584094261835384486, −11.21621060569144140295833142660, −10.31813361393919101976754685542, −9.44805662220618055137708047281, −9.17190297536851853230836775387, −7.90886913406217656999335266726, −7.447795252500097131218416254649, −6.62105010597528422883151980686, −6.29425421975040105157881882480, −5.3754344606164675799654279803, −4.532533753047923291599778071745, −3.62942764410623527508594548843, −2.71951044291899659400328833125, −2.017810965719928082896492756782, −1.147098838429715601376318096764, −0.02002760955301373545872134481, 0.986630586027605592371487776083, 2.397190455276380096185698785803, 3.17713259597812341521207791709, 3.76972382862347789495536169643, 4.48611329157383889635233347634, 5.41166192988443269254138640605, 6.012168875294673212780917851346, 6.636362529882022980195969218436, 7.51848482419623990189683300620, 8.56006709132166384352964338061, 9.155400598073001232450037257891, 9.69461945361040222649322365559, 10.46247045275857409454292421123, 10.94046013755469566000027246964, 11.81864184450185625756546798402, 12.44097823376884307349024389391, 13.13822250362094221145419097423, 13.96206648711495151389640201530, 14.82881423637236179229907758121, 15.20018231273188506741995248691, 15.91255088437381219704681162353, 16.84696809605050669066550921970, 16.869384633140480677714654632420, 17.67233363162906194836660824660, 18.707404047058908285371399195664

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