Properties

Label 1-1436-1436.1435-r0-0-0
Degree $1$
Conductor $1436$
Sign $1$
Analytic cond. $6.66875$
Root an. cond. $6.66875$
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  − 3-s + 5-s + 7-s + 9-s − 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 37-s + 39-s + 41-s + 43-s + 45-s − 47-s + 49-s − 51-s − 53-s − 55-s + ⋯
L(s)  = 1  − 3-s + 5-s + 7-s + 9-s − 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 37-s + 39-s + 41-s + 43-s + 45-s − 47-s + 49-s − 51-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1436 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1436 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1436\)    =    \(2^{2} \cdot 359\)
Sign: $1$
Analytic conductor: \(6.66875\)
Root analytic conductor: \(6.66875\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1436} (1435, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1436,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.481810395\)
\(L(\frac12)\) \(\approx\) \(1.481810395\)
\(L(1)\) \(\approx\) \(1.041718488\)
\(L(1)\) \(\approx\) \(1.041718488\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
359 \( 1 \)
good3 \( 1 - T \)
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

−21.05491217444651484613203822905, −20.20035018852979765655085598102, −18.91506320801277222816481669756, −18.19613112458161981207758941615, −17.78415869913353712149326404416, −17.07864318969196592496815640714, −16.40275443761754175685360569040, −15.549662691084442132445616177610, −14.5422445534566132769263245271, −13.959527154942311854536104177516, −12.9701941917987534194556495938, −12.31096837953157275789414997717, −11.487818800298360713865851819011, −10.74006442528658566730387389585, −9.9176246973379649777448196309, −9.51255207883783691564611278522, −7.92469231841933224369673310845, −7.56545633175025460009756144070, −6.38531211264394885217226832634, −5.42978343987060621122283746670, −5.227895284182962241292828941, −4.241686668135536140722625307000, −2.74373726140052496244780264255, −1.86601111356161335063962543279, −0.88632416021074644498599849723, 0.88632416021074644498599849723, 1.86601111356161335063962543279, 2.74373726140052496244780264255, 4.241686668135536140722625307000, 5.227895284182962241292828941, 5.42978343987060621122283746670, 6.38531211264394885217226832634, 7.56545633175025460009756144070, 7.92469231841933224369673310845, 9.51255207883783691564611278522, 9.9176246973379649777448196309, 10.74006442528658566730387389585, 11.487818800298360713865851819011, 12.31096837953157275789414997717, 12.9701941917987534194556495938, 13.959527154942311854536104177516, 14.5422445534566132769263245271, 15.549662691084442132445616177610, 16.40275443761754175685360569040, 17.07864318969196592496815640714, 17.78415869913353712149326404416, 18.19613112458161981207758941615, 18.91506320801277222816481669756, 20.20035018852979765655085598102, 21.05491217444651484613203822905

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