↑

Properties

Label 1-693-693.535-r0-0-0
Degree $1$
Conductor $693$
Sign $0.287 + 0.957i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.309 + 0.951i)5-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.669 − 0.743i)13-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + 23-s + (−0.809 − 0.587i)25-s + (0.104 + 0.994i)26-s + (0.104 + 0.994i)29-s + (0.978 + 0.207i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.309 + 0.951i)5-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.669 − 0.743i)13-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + 23-s + (−0.809 − 0.587i)25-s + (0.104 + 0.994i)26-s + (0.104 + 0.994i)29-s + (0.978 + 0.207i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7530576039 + 0.5600685502i\)
\(L(\frac12)\) \(\approx\) \(0.7530576039 + 0.5600685502i\)
\(L(1)\) \(\approx\) \(0.7089727243 + 0.3169562731i\)
\(L(1)\) \(\approx\) \(0.7089727243 + 0.3169562731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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

−22.43795507837617083366731453696, −21.32257132330471117512213022991, −20.77771801754676260928901645388, −20.14180021201666295609676037524, −19.23944842624134143231550311728, −18.63709730206700777085535398393, −17.56541862336981316770367133735, −16.955454458473424463133079693471, −16.06899055819565985781820981583, −15.47673261471114067564965741661, −13.81365510070555404846102448860, −13.297654357412168963951426804390, −12.28118621542975225929420918006, −11.61785786689173105279663493005, −10.895834363182616784442717112488, −9.67153915686575349704477653967, −9.067272020676194635940726008526, −8.2967575489204388143546151009, −7.43668627337165394953932726772, −6.2818190638706994049951609896, −4.817858908690202223672253173457, −4.1325891690563057537746621515, −3.01808722358208540745653978594, −1.77024537117419661421377320101, −0.80690030099964166051618052416, 0.930494412491658363960738010994, 2.403334692876937450770283728808, 3.51976859316771701288453020518, 4.83409444864967382028965258078, 5.848821244331323977906368196909, 6.77880785550700036277472119328, 7.36991207370947762839611331795, 8.38740396918714293643934333937, 9.15284029861328466952752120444, 10.32341152975868790822146836154, 10.83733555809504133586804767639, 11.68860068942570556780639085626, 13.16403214304716514334152166402, 13.921645380932565795065203484735, 14.88443969544827059362511025004, 15.47648760595219316041333825303, 16.12362720173939799616069939603, 17.25480095546189919799856658498, 18.065859317866121580761144685379, 18.43545911259076220672665170778, 19.5670272925653825863578371918, 19.97730218620717756509266251551, 21.22495884003337095473737734283, 22.409787028320284488045005283246, 22.86309066232753588030993852246

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