↑

Properties

Label 1-4284-4284.319-r1-0-0
Degree $1$
Conductor $4284$
Sign $0.0281 - 0.999i$
Analytic cond. $460.379$
Root an. cond. $460.379$
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  + i·5-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)19-s − i·23-s − 25-s + (−0.866 − 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.866 − 0.5i)37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 + 0.866i)53-s − 55-s + (−0.5 + 0.866i)59-s + ⋯
L(s)  = 1  + i·5-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)19-s − i·23-s − 25-s + (−0.866 − 0.5i)29-s + (0.866 + 0.5i)31-s + (−0.866 − 0.5i)37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 + 0.866i)53-s − 55-s + (−0.5 + 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4284\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.0281 - 0.999i$
Analytic conductor: \(460.379\)
Root analytic conductor: \(460.379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4284} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4284,\ (1:\ ),\ 0.0281 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1510407896 + 0.1553539375i\)
\(L(\frac12)\) \(\approx\) \(-0.1510407896 + 0.1553539375i\)
\(L(1)\) \(\approx\) \(0.8086092662 + 0.3064555164i\)
\(L(1)\) \(\approx\) \(0.8086092662 + 0.3064555164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good5 \( 1 + iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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

−17.3739937918702046932099497060, −16.97088004460198933868302941951, −16.4816632245248625009801416846, −15.71044492435235958733734338350, −15.0826418557207668563036448369, −14.12260385378190136556386081962, −13.60933733329063282222965976797, −12.95478574357290463209369538269, −12.188858090026529052056680052130, −11.63055446262681667531421605462, −10.89955965096648388866995270398, −10.04449188897108727934546138756, −9.28396612720486618193526439536, −8.57019131882088282504553239004, −8.30465695286433091657163578140, −7.08875414158700631487155085460, −6.56134024153463387618540343804, −5.56868147962743215009040265887, −4.9823627287053887670334091164, −4.24769409538406066041465804993, −3.494960007572922581449054596677, −2.41719349174027658149911340555, −1.69827695319259437345866171884, −0.62319047683198961266393183611, −0.04265464524064523588019646604, 1.38618946874477198871816359738, 2.16264449205835311452307979117, 2.96809979776975080530791558539, 3.670609079611412162428083813588, 4.52920305062512344557569368629, 5.41018624645377078648196574569, 6.14174818437211280769961674003, 6.88481978656257454387095646998, 7.65090844929491855156404887669, 7.996717083192396421091070789117, 9.23566507770709245320944833321, 9.866219447941196780258562233066, 10.44295751033510148393700039095, 11.02277908995941978262297895915, 12.03962982744600036762475066291, 12.36741328157409806901025028063, 13.377730905256340777283784386677, 13.942249037942816369139039187021, 14.891082659053410166307515317901, 15.11874268336206103796219061370, 15.752721577218616212905078328995, 16.87712268734627500583102088933, 17.35843779485836897889627427298, 18.090707770474259746462352382562, 18.51359073344539027753336183943

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