↑

Properties

Label 2-3536-884.883-c0-0-8
Degree $2$
Conductor $3536$
Sign $1$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s + 13-s − 17-s − 2·23-s − 25-s + 4·27-s + 2·39-s + 49-s − 2·51-s − 2·53-s − 4·69-s − 2·75-s − 2·79-s + 5·81-s + 2·101-s − 2·107-s + 3·117-s + ⋯
L(s)  = 1  + 2·3-s + 3·9-s + 13-s − 17-s − 2·23-s − 25-s + 4·27-s + 2·39-s + 49-s − 2·51-s − 2·53-s − 4·69-s − 2·75-s − 2·79-s + 5·81-s + 2·101-s − 2·107-s + 3·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3536} (3535, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.632969686\)
\(L(\frac12)\) \(\approx\) \(2.632969686\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good3 \( ( 1 - T )^{2} \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.656544766899015483700403401215, −8.125191646771596830148438921910, −7.58250572653362020707084047526, −6.69892077643970225054088298112, −5.92003092940087506362364237280, −4.46404156595011367319640004924, −3.98759244338919525738434129546, −3.23935589550066388864693177339, −2.25683189074242146392427305178, −1.61593449884077932397646246437, 1.61593449884077932397646246437, 2.25683189074242146392427305178, 3.23935589550066388864693177339, 3.98759244338919525738434129546, 4.46404156595011367319640004924, 5.92003092940087506362364237280, 6.69892077643970225054088298112, 7.58250572653362020707084047526, 8.125191646771596830148438921910, 8.656544766899015483700403401215

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