Properties

Label 2-232-1.1-c1-0-1
Degree $2$
Conductor $232$
Sign $1$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s − 2·9-s + 3·11-s − 13-s + 15-s + 2·21-s + 4·23-s − 4·25-s − 5·27-s − 29-s + 3·31-s + 3·33-s + 2·35-s − 8·37-s − 39-s − 6·41-s − 5·43-s − 2·45-s + 3·47-s − 3·49-s + 5·53-s + 3·55-s − 8·59-s − 4·63-s − 65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s + 0.436·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 0.185·29-s + 0.538·31-s + 0.522·33-s + 0.338·35-s − 1.31·37-s − 0.160·39-s − 0.937·41-s − 0.762·43-s − 0.298·45-s + 0.437·47-s − 3/7·49-s + 0.686·53-s + 0.404·55-s − 1.04·59-s − 0.503·63-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $1$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594638024\)
\(L(\frac12)\) \(\approx\) \(1.594638024\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.03225617833127401634294644320, −11.35936445667626245776837847935, −10.17646720474260025332202942321, −9.109468260421844793465351652049, −8.430240929203566777866130945662, −7.28385569168970544975317725065, −6.02226938483447033650045844097, −4.83506868338445543793010149058, −3.35801131767532934555280418053, −1.86240791565196501569145297442, 1.86240791565196501569145297442, 3.35801131767532934555280418053, 4.83506868338445543793010149058, 6.02226938483447033650045844097, 7.28385569168970544975317725065, 8.430240929203566777866130945662, 9.109468260421844793465351652049, 10.17646720474260025332202942321, 11.35936445667626245776837847935, 12.03225617833127401634294644320

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