Properties

Label 2-55-55.54-c4-0-13
Degree $2$
Conductor $55$
Sign $1$
Analytic cond. $5.68534$
Root an. cond. $2.38439$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 7·4-s + 25·5-s + 78·7-s − 69·8-s + 81·9-s + 75·10-s + 121·11-s − 162·13-s + 234·14-s − 95·16-s − 402·17-s + 243·18-s − 175·20-s + 363·22-s + 625·25-s − 486·26-s − 546·28-s − 1.59e3·31-s + 819·32-s − 1.20e3·34-s + 1.95e3·35-s − 567·36-s − 1.72e3·40-s − 3.52e3·43-s − 847·44-s + 2.02e3·45-s + ⋯
L(s)  = 1  + 3/4·2-s − 0.437·4-s + 5-s + 1.59·7-s − 1.07·8-s + 9-s + 3/4·10-s + 11-s − 0.958·13-s + 1.19·14-s − 0.371·16-s − 1.39·17-s + 3/4·18-s − 0.437·20-s + 3/4·22-s + 25-s − 0.718·26-s − 0.696·28-s − 1.66·31-s + 0.799·32-s − 1.04·34-s + 1.59·35-s − 0.437·36-s − 1.07·40-s − 1.90·43-s − 0.437·44-s + 45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $1$
Analytic conductor: \(5.68534\)
Root analytic conductor: \(2.38439\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{55} (54, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.512007194\)
\(L(\frac12)\) \(\approx\) \(2.512007194\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p^{2} T \)
11 \( 1 - p^{2} T \)
good2 \( 1 - 3 T + p^{4} T^{2} \)
3 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( 1 - 78 T + p^{4} T^{2} \)
13 \( 1 + 162 T + p^{4} T^{2} \)
17 \( 1 + 402 T + p^{4} T^{2} \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 + 1598 T + p^{4} T^{2} \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 + 3522 T + p^{4} T^{2} \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( 1 - 3442 T + p^{4} T^{2} \)
61 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( 1 + 3998 T + p^{4} T^{2} \)
73 \( 1 - 10638 T + p^{4} T^{2} \)
79 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
83 \( 1 + 13602 T + p^{4} T^{2} \)
89 \( 1 + 15838 T + p^{4} T^{2} \)
97 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
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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

−14.47002882380750157577241342461, −13.54916865829724406339447037671, −12.54094805778707278940313212337, −11.27698924900674473124864967098, −9.766269955181231311435483122057, −8.701121959006908073665828472599, −6.88526828934042534941913164561, −5.23950437555417690129244768773, −4.31669341193265174132537397017, −1.81939565679581751585580500734, 1.81939565679581751585580500734, 4.31669341193265174132537397017, 5.23950437555417690129244768773, 6.88526828934042534941913164561, 8.701121959006908073665828472599, 9.766269955181231311435483122057, 11.27698924900674473124864967098, 12.54094805778707278940313212337, 13.54916865829724406339447037671, 14.47002882380750157577241342461

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