↑

Properties

Label 1-6027-6027.122-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.253 + 0.967i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.0747 + 0.997i)11-s + (−0.900 − 0.433i)13-s + (−0.733 − 0.680i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)20-s + (−0.623 − 0.781i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (−0.955 + 0.294i)10-s + (0.0747 + 0.997i)11-s + (−0.900 − 0.433i)13-s + (−0.733 − 0.680i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)20-s + (−0.623 − 0.781i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.253 + 0.967i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.253 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8997020795 + 1.166051510i\)
\(L(\frac12)\) \(\approx\) \(0.8997020795 + 1.166051510i\)
\(L(1)\) \(\approx\) \(0.8341144071 + 0.3638458279i\)
\(L(1)\) \(\approx\) \(0.8341144071 + 0.3638458279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + T \)
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

−17.44251987051150467608709448667, −16.88946696824870457703264171090, −16.637742731530060458444824358128, −15.798295669726982704672442100187, −14.87562000351848131326537114564, −14.12565460679425287880062231179, −13.43481620958763229412042022273, −12.86961127787629754571078734205, −12.156127616722919274758309927428, −11.50338155400494364961753424380, −10.77647326094698325617713498438, −10.17095360085516795290579231825, −9.52414667495262116616110516178, −8.92252944346890853261517040931, −8.47274441499338060185721688269, −7.47507534786354929879599990830, −6.884433288638631638985886393286, −6.10346616944633202689686890920, −5.27129666164908868811057139991, −4.5316161674165221927501392541, −3.50643474554037685262724277416, −2.73553635408375229057842466674, −2.21075415059925222201853873168, −1.222466571841359045907225879448, −0.5684666003478400184401098259, 0.9905021046534392831590483936, 1.69536851971684114153420065231, 2.46467738762083471394066124292, 3.16895025535490099972087690351, 4.60304496133366116224354623466, 5.106414650990013546099997725783, 5.86464021676143327786069002239, 6.590944902035647487924942387607, 7.062325489596117687642167620714, 7.933046105309065517411636664495, 8.42420936226223505413581064063, 9.474580011855090537656105412239, 9.9674126830156950191167001475, 10.16153893014617670363012636967, 11.052272312797942893769387850702, 11.95448634178272809369752753722, 12.652221202071338206467244483358, 13.37022052953122773561270198608, 14.47331566681240764336652650359, 14.54234106701697370646551080188, 15.19468981007318988990808151423, 16.07074502400888404956398973392, 16.85569202334128180144998211865, 17.26201348100363809393249590368, 17.749896113225349688252596104922

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