Properties

Label 2-55-1.1-c3-0-9
Degree $2$
Conductor $55$
Sign $-1$
Analytic cond. $3.24510$
Root an. cond. $1.80141$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 7·4-s − 5·5-s − 3·6-s − 9·7-s − 15·8-s − 18·9-s − 5·10-s + 11·11-s + 21·12-s + 2·13-s − 9·14-s + 15·15-s + 41·16-s + 21·17-s − 18·18-s − 85·19-s + 35·20-s + 27·21-s + 11·22-s + 22·23-s + 45·24-s + 25·25-s + 2·26-s + 135·27-s + 63·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s − 0.204·6-s − 0.485·7-s − 0.662·8-s − 2/3·9-s − 0.158·10-s + 0.301·11-s + 0.505·12-s + 0.0426·13-s − 0.171·14-s + 0.258·15-s + 0.640·16-s + 0.299·17-s − 0.235·18-s − 1.02·19-s + 0.391·20-s + 0.280·21-s + 0.106·22-s + 0.199·23-s + 0.382·24-s + 1/5·25-s + 0.0150·26-s + 0.962·27-s + 0.425·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(3.24510\)
Root analytic conductor: \(1.80141\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
11 \( 1 - p T \)
good2 \( 1 - T + p^{3} T^{2} \)
3 \( 1 + p T + p^{3} T^{2} \)
7 \( 1 + 9 T + p^{3} T^{2} \)
13 \( 1 - 2 T + p^{3} T^{2} \)
17 \( 1 - 21 T + p^{3} T^{2} \)
19 \( 1 + 85 T + p^{3} T^{2} \)
23 \( 1 - 22 T + p^{3} T^{2} \)
29 \( 1 + 165 T + p^{3} T^{2} \)
31 \( 1 + 83 T + p^{3} T^{2} \)
37 \( 1 - T + p^{3} T^{2} \)
41 \( 1 + 478 T + p^{3} T^{2} \)
43 \( 1 + 8 T + p^{3} T^{2} \)
47 \( 1 - 126 T + p^{3} T^{2} \)
53 \( 1 + 683 T + p^{3} T^{2} \)
59 \( 1 + 290 T + p^{3} T^{2} \)
61 \( 1 - 257 T + p^{3} T^{2} \)
67 \( 1 - 776 T + p^{3} T^{2} \)
71 \( 1 + 313 T + p^{3} T^{2} \)
73 \( 1 - 902 T + p^{3} T^{2} \)
79 \( 1 - 830 T + p^{3} T^{2} \)
83 \( 1 - 842 T + p^{3} T^{2} \)
89 \( 1 - 25 T + p^{3} T^{2} \)
97 \( 1 + 1784 T + p^{3} T^{2} \)
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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

−14.23818884762054503300538549627, −13.01744979915157662584740759019, −12.11542215936679308087649861172, −10.91863500584103319510480633270, −9.440842169835386193433429604798, −8.273305932618668249050131927195, −6.40740806035650295877386950202, −5.10483814859930842390914923141, −3.57122137945214328716009661090, 0, 3.57122137945214328716009661090, 5.10483814859930842390914923141, 6.40740806035650295877386950202, 8.273305932618668249050131927195, 9.440842169835386193433429604798, 10.91863500584103319510480633270, 12.11542215936679308087649861172, 13.01744979915157662584740759019, 14.23818884762054503300538549627

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