↑

Properties

Label 1-1205-1205.953-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.537 - 0.843i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.5i)4-s − i·6-s + (−0.608 + 0.793i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 + 0.608i)11-s + (−0.258 + 0.965i)12-s + (−0.608 − 0.793i)13-s + (0.793 − 0.608i)14-s + (0.5 + 0.866i)16-s + (0.382 − 0.923i)17-s + (0.965 − 0.258i)18-s + (0.608 − 0.793i)19-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.5i)4-s − i·6-s + (−0.608 + 0.793i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.793 + 0.608i)11-s + (−0.258 + 0.965i)12-s + (−0.608 − 0.793i)13-s + (0.793 − 0.608i)14-s + (0.5 + 0.866i)16-s + (0.382 − 0.923i)17-s + (0.965 − 0.258i)18-s + (0.608 − 0.793i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4808249377 - 0.2638810385i\)
\(L(\frac12)\) \(\approx\) \(0.4808249377 - 0.2638810385i\)
\(L(1)\) \(\approx\) \(0.6265765971 + 0.08869347498i\)
\(L(1)\) \(\approx\) \(0.6265765971 + 0.08869347498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (-0.608 + 0.793i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (-0.608 - 0.793i)T \)
17 \( 1 + (0.382 - 0.923i)T \)
19 \( 1 + (0.608 - 0.793i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (0.130 + 0.991i)T \)
37 \( 1 + (0.608 - 0.793i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (-0.130 - 0.991i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.130 - 0.991i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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

−21.02546306371944622952807759897, −19.97393126321884329401627467671, −19.73210722801525480296143314987, −18.90662093491247243816204387270, −18.46717897480055446741943715161, −17.29969576752416891125739249522, −16.86858726085472409955083664240, −16.32957251344201537071556880160, −15.01286566329758681838230400563, −14.377223212081888739088301246806, −13.633240338663740792203009316807, −12.701016062355261000713388605, −11.68385373864080455042010911710, −11.27659122513056253553975199796, −9.851794351611065402607023673850, −9.5736897898668968132555518711, −8.40086165151971084012309254716, −7.79387063037358328379516450690, −6.97714219047619519592995272821, −6.356244467815806928548204362406, −5.61630571141578118299085542611, −3.87324612044284763511732551606, −3.04813680913458712454306817307, −1.737499888365586420282221931318, −1.16291640568615245591089296737, 0.307301369197551272196488813350, 2.02254342906574475989135016343, 2.86518591423049267126572118691, 3.51957487721744536588231576060, 4.79652885628991421638653066646, 5.70764284120105094151902259231, 6.80130249635304823299556732400, 7.64810785117463871090404191008, 8.721132827192132921711017895292, 9.25625370772838756154293377386, 9.88106590706294152808229725139, 10.51848648349568101784425463212, 11.66319006101819957860472185706, 12.08536109393312259378105400544, 13.10974202101923909175938743914, 14.40239584522500758122553172724, 15.13185802190479402513123025586, 15.75630143357671639506411247549, 16.44872512154461945724636959862, 17.16566260851238036894740808820, 18.01465888989366493939566758993, 18.75508000752949913211043892999, 19.8508884022580435389013652037, 20.00186575160725261053720303980, 20.855367572142245955157322411515

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