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Properties

Label 1-12e3-1728.1301-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.980 - 0.198i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (0.300 + 0.953i)5-s + (−0.906 − 0.422i)7-s + (−0.461 − 0.887i)11-s + (−0.843 + 0.537i)13-s + (0.866 − 0.5i)17-s + (0.991 + 0.130i)19-s + (−0.906 + 0.422i)23-s + (−0.819 + 0.573i)25-s + (0.216 + 0.976i)29-s + (0.939 + 0.342i)31-s + (0.130 − 0.991i)35-s + (0.991 − 0.130i)37-s + (0.819 + 0.573i)41-s + (−0.461 − 0.887i)43-s + (−0.342 − 0.939i)47-s + ⋯
L(s)  = 1  + (0.300 + 0.953i)5-s + (−0.906 − 0.422i)7-s + (−0.461 − 0.887i)11-s + (−0.843 + 0.537i)13-s + (0.866 − 0.5i)17-s + (0.991 + 0.130i)19-s + (−0.906 + 0.422i)23-s + (−0.819 + 0.573i)25-s + (0.216 + 0.976i)29-s + (0.939 + 0.342i)31-s + (0.130 − 0.991i)35-s + (0.991 − 0.130i)37-s + (0.819 + 0.573i)41-s + (−0.461 − 0.887i)43-s + (−0.342 − 0.939i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.980 - 0.198i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.980 - 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02178707098 + 0.2171596822i\)
\(L(\frac12)\) \(\approx\) \(0.02178707098 + 0.2171596822i\)
\(L(1)\) \(\approx\) \(0.8676685930 + 0.1166894692i\)
\(L(1)\) \(\approx\) \(0.8676685930 + 0.1166894692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.300 + 0.953i)T \)
7 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (-0.461 - 0.887i)T \)
13 \( 1 + (-0.843 + 0.537i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.991 + 0.130i)T \)
23 \( 1 + (-0.906 + 0.422i)T \)
29 \( 1 + (0.216 + 0.976i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (0.819 + 0.573i)T \)
43 \( 1 + (-0.461 - 0.887i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.300 + 0.953i)T \)
61 \( 1 + (0.737 - 0.675i)T \)
67 \( 1 + (-0.537 - 0.843i)T \)
71 \( 1 + (-0.258 + 0.965i)T \)
73 \( 1 + (0.258 + 0.965i)T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (0.976 - 0.216i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−19.67054075753763159313991117128, −19.18525625751116504613750129920, −18.054563010442264904129598675966, −17.56498571216427554998828103799, −16.68345720131899272462705346196, −16.049495041596865538369208768983, −15.39784797100226904583258127344, −14.55493821858089748134483847955, −13.57617286283727279590333375209, −12.83795246699403189666048947897, −12.34130037509655191377040392141, −11.76463754717974978190340089812, −10.30582118753926872539660934930, −9.73774076828558864321902767955, −9.37175688853399826935716014520, −8.04069616963350215454012876391, −7.72876390109469745896565762865, −6.40834664318302046144247166692, −5.74075077208723289067217264484, −4.94536281705358336420927011206, −4.15733448165791062037797684646, −2.93626643983198003901247731438, −2.23710171266594532580533245416, −1.0306867392191821059841345751, −0.04526519157942563631836946571, 1.10529789876729449218399481387, 2.47092496690153607695963097645, 3.12251628523862814819110450769, 3.795898847647222148349094250089, 5.11694289352254985444406276251, 5.88716197401938764626967169952, 6.71256798635630775925365556764, 7.36128604366684272625193546133, 8.137209053161696803052998795390, 9.47130571514413190765718664000, 9.85973124446155831286864113702, 10.57343537931954807635618150946, 11.49363646212730015179009269004, 12.16669264368466407332549245791, 13.19488626786401946691914929150, 13.98122554936545037638370581134, 14.26238222568250393929203229390, 15.34904183517836363787766938532, 16.241565190760582469919260783391, 16.57288557095081854281869532782, 17.68600501244556247305566629014, 18.35508701881299948172963470175, 19.031181195271070840797462285917, 19.60617412254452313854401798739, 20.42368467433456732668878180489

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