Properties

Label 2-55-55.54-c0-0-0
Degree $2$
Conductor $55$
Sign $1$
Analytic cond. $0.0274485$
Root an. cond. $0.165676$
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 + 16-s + 20-s + 25-s − 2·31-s − 36-s + 44-s − 45-s − 49-s + 55-s + 2·59-s − 64-s + 2·71-s − 80-s + 81-s − 2·89-s − 99-s − 100-s + ⋯
L(s)  = 1  − 4-s − 5-s + 9-s − 11-s + 16-s + 20-s + 25-s − 2·31-s − 36-s + 44-s − 45-s − 49-s + 55-s + 2·59-s − 64-s + 2·71-s − 80-s + 81-s − 2·89-s − 99-s − 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.0274485\)
Root analytic conductor: \(0.165676\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{55} (54, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3736291621\)
\(L(\frac12)\) \(\approx\) \(0.3736291621\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + T^{2} \)
3 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 + T )^{2} \)
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

−15.59186099364208642110354093174, −14.56717603366310796920991740421, −13.16149603660972941306714478301, −12.50918611307147520131121898334, −10.94513953936249769419384576241, −9.733624443992794679615252048198, −8.364696256624625196471110219171, −7.30588121583844319826286747084, −5.10842423248229864558432670603, −3.81577785438201065364975996334, 3.81577785438201065364975996334, 5.10842423248229864558432670603, 7.30588121583844319826286747084, 8.364696256624625196471110219171, 9.733624443992794679615252048198, 10.94513953936249769419384576241, 12.50918611307147520131121898334, 13.16149603660972941306714478301, 14.56717603366310796920991740421, 15.59186099364208642110354093174

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