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Properties

Label 1-4851-4851.2279-r0-0-0
Degree $1$
Conductor $4851$
Sign $0.680 + 0.733i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
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  + (0.473 + 0.880i)2-s + (−0.550 + 0.834i)4-s + (−0.193 + 0.981i)5-s + (−0.995 − 0.0896i)8-s + (−0.955 + 0.294i)10-s + (0.999 + 0.0299i)13-s + (−0.393 − 0.919i)16-s + (0.887 − 0.460i)17-s + (−0.669 − 0.743i)19-s + (−0.712 − 0.701i)20-s + (−0.365 + 0.930i)23-s + (−0.925 − 0.379i)25-s + (0.447 + 0.894i)26-s + (−0.163 − 0.986i)29-s + (−0.809 − 0.587i)31-s + (0.623 − 0.781i)32-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + (−0.550 + 0.834i)4-s + (−0.193 + 0.981i)5-s + (−0.995 − 0.0896i)8-s + (−0.955 + 0.294i)10-s + (0.999 + 0.0299i)13-s + (−0.393 − 0.919i)16-s + (0.887 − 0.460i)17-s + (−0.669 − 0.743i)19-s + (−0.712 − 0.701i)20-s + (−0.365 + 0.930i)23-s + (−0.925 − 0.379i)25-s + (0.447 + 0.894i)26-s + (−0.163 − 0.986i)29-s + (−0.809 − 0.587i)31-s + (0.623 − 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.680 + 0.733i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (2279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ 0.680 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540185424 + 0.6720659212i\)
\(L(\frac12)\) \(\approx\) \(1.540185424 + 0.6720659212i\)
\(L(1)\) \(\approx\) \(0.9970571933 + 0.6341859133i\)
\(L(1)\) \(\approx\) \(0.9970571933 + 0.6341859133i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (-0.193 + 0.981i)T \)
13 \( 1 + (0.999 + 0.0299i)T \)
17 \( 1 + (0.887 - 0.460i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (-0.163 - 0.986i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.772 + 0.635i)T \)
41 \( 1 + (0.575 - 0.817i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (0.691 - 0.722i)T \)
53 \( 1 + (0.842 - 0.538i)T \)
59 \( 1 + (0.995 - 0.0896i)T \)
61 \( 1 + (0.0448 - 0.998i)T \)
67 \( 1 + T \)
71 \( 1 + (0.963 + 0.266i)T \)
73 \( 1 + (0.280 - 0.959i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.999 + 0.0299i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−18.28102675803324185561586148929, −17.42200933629841906427393832675, −16.5456950206528551962974241580, −16.13601099096259960318881744888, −15.22382468813756319981694135921, −14.489496732426851220893026419780, −13.96349604335104858844320322732, −13.060045369121561434070130750558, −12.602483570304027128390961925914, −12.18370194826089498982515859008, −11.32892339578274558285108703606, −10.60973201930645939024936671143, −10.100298125988374878189013603333, −9.14898170246230895042813628885, −8.60087690766659326925337895593, −8.07488005067012907880967918149, −6.87990786860239907969730960279, −5.8860929607834318543097617434, −5.50703842315566842342631270447, −4.63155462137414489820927412615, −3.8597002781951821975193196122, −3.48442905926591181393000645023, −2.29300358573214657037804578479, −1.490154466615040637937171712809, −0.89824225998287676731731515855, 0.456914005830096446536876078589, 1.973142046691880484525933824235, 2.84881513122769475275006148581, 3.73279154845056557110874962278, 3.97422309434584981895690507322, 5.25535768428336628245230476425, 5.715020219948128383794225418957, 6.6247831566479904032941749431, 6.990933861032920290899891343761, 7.87827767888367668309095352338, 8.32939631960426817085331512562, 9.31737144265323060361269575306, 9.93470443260168353293521032200, 10.9289247077728904059966305032, 11.51169239534488876900865643785, 12.16430162153595491056563560367, 13.106618966869900567311399387757, 13.71858106316759171800573199074, 14.12521820537112231386844598512, 15.1126622738137468994641837881, 15.3083200898320246680601152217, 16.06700356284421560615498838119, 16.77184356931491575600424982549, 17.50710333496519406725334649336, 18.099832363215076630795036265770

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