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Properties

Label 1-3639-3639.3638-r1-0-0
Degree $1$
Conductor $3639$
Sign $1$
Analytic cond. $391.064$
Root an. cond. $391.064$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 17-s + 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s − 29-s + 31-s + 32-s + 34-s + 35-s − 37-s + 38-s + 40-s + 41-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 17-s + 19-s + 20-s − 22-s + 23-s + 25-s + 26-s + 28-s − 29-s + 31-s + 32-s + 34-s + 35-s − 37-s + 38-s + 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3639\)    =    \(3 \cdot 1213\)
Sign: $1$
Analytic conductor: \(391.064\)
Root analytic conductor: \(391.064\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3639} (3638, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3639,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(9.459656507\)
\(L(\frac12)\) \(\approx\) \(9.459656507\)
\(L(1)\) \(\approx\) \(3.124712719\)
\(L(1)\) \(\approx\) \(3.124712719\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
1213 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−18.58367879963017341357726497125, −17.65975386838780357008632635694, −17.19299054358472136601905592539, −16.22811105673769111605208905376, −15.73538741213355115146938503015, −14.836971702598716739506502322298, −14.298333606968590289537018190713, −13.62803600983229710395994895807, −13.21525362218677645493071656417, −12.38575097564730684374943552579, −11.63305584907437697818696165700, −10.78624578464416036903737955891, −10.505530216429790845543498492744, −9.48766869039830463576989262959, −8.57124947627019343104415263335, −7.67220210796016327740310163852, −7.187213316885645640032481508090, −6.01042202250929038614164127913, −5.57185105760512712570554615161, −5.04613516166018393769423155517, −4.189680716492999041786263325320, −3.123217032284718087922604209343, −2.58830403525562177116330423003, −1.52843900833230563777418215779, −1.07464122690799140819291446455, 1.07464122690799140819291446455, 1.52843900833230563777418215779, 2.58830403525562177116330423003, 3.123217032284718087922604209343, 4.189680716492999041786263325320, 5.04613516166018393769423155517, 5.57185105760512712570554615161, 6.01042202250929038614164127913, 7.187213316885645640032481508090, 7.67220210796016327740310163852, 8.57124947627019343104415263335, 9.48766869039830463576989262959, 10.505530216429790845543498492744, 10.78624578464416036903737955891, 11.63305584907437697818696165700, 12.38575097564730684374943552579, 13.21525362218677645493071656417, 13.62803600983229710395994895807, 14.298333606968590289537018190713, 14.836971702598716739506502322298, 15.73538741213355115146938503015, 16.22811105673769111605208905376, 17.19299054358472136601905592539, 17.65975386838780357008632635694, 18.58367879963017341357726497125

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