↑

Properties

Label 2-212-212.211-c0-0-1
Degree $2$
Conductor $212$
Sign $1$
Analytic cond. $0.105801$
Root an. cond. $0.325271$
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-s − 3-s + 4-s − 6-s + 8-s − 12-s − 13-s + 16-s − 17-s − 19-s − 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 2·31-s + 32-s − 34-s − 37-s − 38-s + 39-s − 46-s − 48-s + 49-s + 50-s + 51-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s − 12-s − 13-s + 16-s − 17-s − 19-s − 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 2·31-s + 32-s − 34-s − 37-s − 38-s + 39-s − 46-s − 48-s + 49-s + 50-s + 51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $1$
Analytic conductor: \(0.105801\)
Root analytic conductor: \(0.325271\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{212} (211, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 212,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8980697988\)
\(L(\frac12)\) \(\approx\) \(0.8980697988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.41750948045171590969934947186, −11.82901665658109116724648691725, −10.91627553799841524873777369470, −10.14445238105907303641926611318, −8.481483024997906207064325219729, −7.04796772458452462662713597640, −6.25028881503830103293354510395, −5.19875632721977887819430442161, −4.26287604595125770225335555293, −2.46540880625352001412354028566, 2.46540880625352001412354028566, 4.26287604595125770225335555293, 5.19875632721977887819430442161, 6.25028881503830103293354510395, 7.04796772458452462662713597640, 8.481483024997906207064325219729, 10.14445238105907303641926611318, 10.91627553799841524873777369470, 11.82901665658109116724648691725, 12.41750948045171590969934947186

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