Properties

Label 1-18e2-324.311-r0-0-0
Degree $1$
Conductor $324$
Sign $0.778 - 0.627i$
Analytic cond. $1.50464$
Root an. cond. $1.50464$
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  + (0.835 − 0.549i)5-s + (0.686 − 0.727i)7-s + (0.893 − 0.448i)11-s + (0.396 + 0.918i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.686 − 0.727i)23-s + (0.396 − 0.918i)25-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (0.173 − 0.984i)35-s + (0.173 + 0.984i)37-s + (−0.597 + 0.802i)41-s + (0.0581 − 0.998i)43-s + (0.973 + 0.230i)47-s + ⋯
L(s)  = 1  + (0.835 − 0.549i)5-s + (0.686 − 0.727i)7-s + (0.893 − 0.448i)11-s + (0.396 + 0.918i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.686 − 0.727i)23-s + (0.396 − 0.918i)25-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (0.173 − 0.984i)35-s + (0.173 + 0.984i)37-s + (−0.597 + 0.802i)41-s + (0.0581 − 0.998i)43-s + (0.973 + 0.230i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ 0.778 - 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.506826601 - 0.5319515718i\)
\(L(\frac12)\) \(\approx\) \(1.506826601 - 0.5319515718i\)
\(L(1)\) \(\approx\) \(1.288300030 - 0.2366059094i\)
\(L(1)\) \(\approx\) \(1.288300030 - 0.2366059094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.835 - 0.549i)T \)
7 \( 1 + (0.686 - 0.727i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (0.396 + 0.918i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.686 - 0.727i)T \)
29 \( 1 + (0.993 - 0.116i)T \)
31 \( 1 + (-0.973 + 0.230i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.597 + 0.802i)T \)
43 \( 1 + (0.0581 - 0.998i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.893 + 0.448i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.835 - 0.549i)T \)
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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

−25.19533658111480774505180228976, −24.601516124664017713938962123127, −23.31024801551567494611733415046, −22.36283882308560522656473919630, −21.748284536993771019552349307209, −20.83621219845349062327425414786, −19.9141970395527942363715659936, −18.75090011420682117978381868232, −17.80203451911603949584602115855, −17.5274395059005486706742836973, −16.09097222826075324828355114762, −15.009405251152614775058951631564, −14.41219724094856715540392404699, −13.40398373549580413914928613612, −12.32266048348768160067818722461, −11.3283372523452427166889901102, −10.39340412911126014428009513477, −9.361176064272551665022702524560, −8.46876428042493918272234423340, −7.20698937241726350475286210260, −6.11798676049396337490991720299, −5.32431383966231684339561717035, −3.94980129439190817044106517934, −2.53184910001294917870214585648, −1.62487431203904762012306271048, 1.19175835911422244249005533681, 2.170879577511768091912796196703, 4.005173954634634214560267561564, 4.70214708226528584286890147111, 6.1294584982910593812733501864, 6.83735017929738893887704935032, 8.44003707791148259835873932072, 8.95036673988890943919045229985, 10.227687987842018034435022848074, 11.09052560930006375260721137983, 12.12911170407009296871521177154, 13.3505795713996735222353489333, 13.94310601948264034867677710330, 14.80873435091652780556362051341, 16.214298198980763351538366643237, 17.02090230501064952825142385198, 17.562621542864341571106754904256, 18.67950943654851233176067627891, 19.85315599383210676713796886982, 20.49788436347581387110983562136, 21.617800193245869612144797200643, 21.96173089901597853872926838304, 23.56628468378552387997121732965, 24.04167575894162826737926228712, 24.92719847890771581607227458135

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