Properties

Label 2-211-211.210-c0-0-0
Degree $2$
Conductor $211$
Sign $1$
Analytic cond. $0.105302$
Root an. cond. $0.324503$
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  + 4-s − 5-s + 9-s − 11-s − 13-s + 16-s − 19-s − 20-s + 36-s − 37-s − 43-s − 44-s − 45-s − 47-s + 49-s − 52-s + 2·53-s + 55-s + 2·59-s + 64-s + 65-s − 71-s + 2·73-s − 76-s − 79-s − 80-s + 81-s + ⋯
L(s)  = 1  + 4-s − 5-s + 9-s − 11-s − 13-s + 16-s − 19-s − 20-s + 36-s − 37-s − 43-s − 44-s − 45-s − 47-s + 49-s − 52-s + 2·53-s + 55-s + 2·59-s + 64-s + 65-s − 71-s + 2·73-s − 76-s − 79-s − 80-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(211\)
Sign: $1$
Analytic conductor: \(0.105302\)
Root analytic conductor: \(0.324503\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{211} (210, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 211,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7358031812\)
\(L(\frac12)\) \(\approx\) \(0.7358031812\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39814787958180065980360115033, −11.74104219053413987752461018312, −10.64078657438421046185276150704, −10.01038163356747423234171091982, −8.324932616794407432977756050213, −7.46386740565527729315092106349, −6.77516729846439138358216471580, −5.18037386937466737132796711185, −3.83733924492078633030089718140, −2.29906709754737556376540443181, 2.29906709754737556376540443181, 3.83733924492078633030089718140, 5.18037386937466737132796711185, 6.77516729846439138358216471580, 7.46386740565527729315092106349, 8.324932616794407432977756050213, 10.01038163356747423234171091982, 10.64078657438421046185276150704, 11.74104219053413987752461018312, 12.39814787958180065980360115033

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