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Properties

Label 1-12e3-1728.1051-r1-0-0
Degree $1$
Conductor $1728$
Sign $0.503 + 0.864i$
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  + (−0.537 + 0.843i)5-s + (0.996 + 0.0871i)7-s + (−0.976 − 0.216i)11-s + (0.737 − 0.675i)13-s + (0.866 + 0.5i)17-s + (0.793 + 0.608i)19-s + (−0.996 + 0.0871i)23-s + (−0.422 − 0.906i)25-s + (−0.0436 − 0.999i)29-s + (0.766 + 0.642i)31-s + (−0.608 + 0.793i)35-s + (−0.793 + 0.608i)37-s + (−0.422 + 0.906i)41-s + (0.976 + 0.216i)43-s + (0.642 + 0.766i)47-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)5-s + (0.996 + 0.0871i)7-s + (−0.976 − 0.216i)11-s + (0.737 − 0.675i)13-s + (0.866 + 0.5i)17-s + (0.793 + 0.608i)19-s + (−0.996 + 0.0871i)23-s + (−0.422 − 0.906i)25-s + (−0.0436 − 0.999i)29-s + (0.766 + 0.642i)31-s + (−0.608 + 0.793i)35-s + (−0.793 + 0.608i)37-s + (−0.422 + 0.906i)41-s + (0.976 + 0.216i)43-s + (0.642 + 0.766i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ 0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.911703233 + 1.099093131i\)
\(L(\frac12)\) \(\approx\) \(1.911703233 + 1.099093131i\)
\(L(1)\) \(\approx\) \(1.110292367 + 0.2084153060i\)
\(L(1)\) \(\approx\) \(1.110292367 + 0.2084153060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.537 + 0.843i)T \)
7 \( 1 + (0.996 + 0.0871i)T \)
11 \( 1 + (-0.976 - 0.216i)T \)
13 \( 1 + (0.737 - 0.675i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.793 + 0.608i)T \)
23 \( 1 + (-0.996 + 0.0871i)T \)
29 \( 1 + (-0.0436 - 0.999i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.793 + 0.608i)T \)
41 \( 1 + (-0.422 + 0.906i)T \)
43 \( 1 + (0.976 + 0.216i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.537 - 0.843i)T \)
61 \( 1 + (-0.887 - 0.461i)T \)
67 \( 1 + (-0.675 - 0.737i)T \)
71 \( 1 + (-0.965 + 0.258i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (-0.342 + 0.939i)T \)
83 \( 1 + (0.999 - 0.0436i)T \)
89 \( 1 + (0.258 - 0.965i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−20.218744247775186030255127367772, −19.23622194767667158005493471915, −18.41544265322681588170390337775, −17.87439769576623440510843109106, −16.977573272742342022004940889385, −16.14646655772992323554833405728, −15.73761098731474140949427122060, −14.84258129900242979976866419769, −13.819770031943927021576478897540, −13.48081878948387959828915797228, −12.148283942904491859205235747651, −11.97788055923323713897887129322, −10.97441592693776825868348909531, −10.26172163949914452590910816084, −9.11597884255465350858118229322, −8.59193811789995570220412202090, −7.64413941586411322546185151531, −7.29820542843154680711800902009, −5.78151087501088500352127102085, −5.17161975306088688485474044855, −4.42134867413482153373639218743, −3.61468836576233299671804032370, −2.38740654690426544117100916622, −1.38570698564510470403466316253, −0.54593539222931625418777415348, 0.78051813717748220012419715530, 1.86687871542556715304179484986, 2.952310084611998984728683796433, 3.600367709577532554905282309, 4.62087365809055419859459942709, 5.606279355428017134413543102623, 6.20073549753459211452451389139, 7.48908897008876424360265055499, 7.98729294584612636485979663345, 8.415316088508324718657083550993, 9.95690124193959035167630710151, 10.40858036667836433396631752310, 11.20649707788413662946988059244, 11.85056000957881111803977065478, 12.634039768108574779542332288457, 13.82417097100901313977594725628, 14.15594081034247986633984058148, 15.1883186766339317656560194015, 15.602712422724012742701808612537, 16.390990301240634877397798283882, 17.54105521333840977339136482110, 18.08360403568112407921183682694, 18.66677743347987080025743973310, 19.33787621312211653556757187510, 20.38917937833109594915243655967

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