Properties

Label 2-22-1.1-c5-0-0
Degree $2$
Conductor $22$
Sign $1$
Analytic cond. $3.52844$
Root an. cond. $1.87841$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 21·3-s + 16·4-s + 81·5-s + 84·6-s + 98·7-s − 64·8-s + 198·9-s − 324·10-s + 121·11-s − 336·12-s + 824·13-s − 392·14-s − 1.70e3·15-s + 256·16-s + 978·17-s − 792·18-s − 2.14e3·19-s + 1.29e3·20-s − 2.05e3·21-s − 484·22-s + 3.69e3·23-s + 1.34e3·24-s + 3.43e3·25-s − 3.29e3·26-s + 945·27-s + 1.56e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·3-s + 1/2·4-s + 1.44·5-s + 0.952·6-s + 0.755·7-s − 0.353·8-s + 0.814·9-s − 1.02·10-s + 0.301·11-s − 0.673·12-s + 1.35·13-s − 0.534·14-s − 1.95·15-s + 1/4·16-s + 0.820·17-s − 0.576·18-s − 1.35·19-s + 0.724·20-s − 1.01·21-s − 0.213·22-s + 1.45·23-s + 0.476·24-s + 1.09·25-s − 0.956·26-s + 0.249·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $1$
Analytic conductor: \(3.52844\)
Root analytic conductor: \(1.87841\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9406718784\)
\(L(\frac12)\) \(\approx\) \(0.9406718784\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
11 \( 1 - p^{2} T \)
good3 \( 1 + 7 p T + p^{5} T^{2} \)
5 \( 1 - 81 T + p^{5} T^{2} \)
7 \( 1 - 2 p^{2} T + p^{5} T^{2} \)
13 \( 1 - 824 T + p^{5} T^{2} \)
17 \( 1 - 978 T + p^{5} T^{2} \)
19 \( 1 + 2140 T + p^{5} T^{2} \)
23 \( 1 - 3699 T + p^{5} T^{2} \)
29 \( 1 - 120 p T + p^{5} T^{2} \)
31 \( 1 + 7813 T + p^{5} T^{2} \)
37 \( 1 + 13597 T + p^{5} T^{2} \)
41 \( 1 - 6492 T + p^{5} T^{2} \)
43 \( 1 - 14234 T + p^{5} T^{2} \)
47 \( 1 + 20352 T + p^{5} T^{2} \)
53 \( 1 + 366 T + p^{5} T^{2} \)
59 \( 1 - 9825 T + p^{5} T^{2} \)
61 \( 1 - 26132 T + p^{5} T^{2} \)
67 \( 1 - 17093 T + p^{5} T^{2} \)
71 \( 1 + 23583 T + p^{5} T^{2} \)
73 \( 1 + 35176 T + p^{5} T^{2} \)
79 \( 1 + 42490 T + p^{5} T^{2} \)
83 \( 1 - 22674 T + p^{5} T^{2} \)
89 \( 1 + 17145 T + p^{5} T^{2} \)
97 \( 1 + 30727 T + p^{5} T^{2} \)
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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

−17.32068231276180834664516858820, −16.29548411569198450994408504892, −14.45847546651427787147018449759, −12.83217645361744585731682355329, −11.22757978927652857014871325453, −10.44211558850650432678746479523, −8.821410682964445825677633750722, −6.54068594782954764725318890123, −5.43962434674073784085121528752, −1.38413119829197160596611746929, 1.38413119829197160596611746929, 5.43962434674073784085121528752, 6.54068594782954764725318890123, 8.821410682964445825677633750722, 10.44211558850650432678746479523, 11.22757978927652857014871325453, 12.83217645361744585731682355329, 14.45847546651427787147018449759, 16.29548411569198450994408504892, 17.32068231276180834664516858820

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