Properties

Label 2-231-1.1-c1-0-6
Degree $2$
Conductor $231$
Sign $1$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.79·2-s − 3-s + 1.20·4-s + 3·5-s − 1.79·6-s + 7-s − 1.41·8-s + 9-s + 5.37·10-s − 11-s − 1.20·12-s + 13-s + 1.79·14-s − 3·15-s − 4.95·16-s + 7.58·17-s + 1.79·18-s − 6.58·19-s + 3.62·20-s − 21-s − 1.79·22-s − 5.58·23-s + 1.41·24-s + 4·25-s + 1.79·26-s − 27-s + 1.20·28-s + ⋯
L(s)  = 1  + 1.26·2-s − 0.577·3-s + 0.604·4-s + 1.34·5-s − 0.731·6-s + 0.377·7-s − 0.501·8-s + 0.333·9-s + 1.69·10-s − 0.301·11-s − 0.348·12-s + 0.277·13-s + 0.478·14-s − 0.774·15-s − 1.23·16-s + 1.83·17-s + 0.422·18-s − 1.51·19-s + 0.810·20-s − 0.218·21-s − 0.381·22-s − 1.16·23-s + 0.289·24-s + 0.800·25-s + 0.351·26-s − 0.192·27-s + 0.228·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.153673043\)
\(L(\frac12)\) \(\approx\) \(2.153673043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 - 1.79T + 2T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
19 \( 1 + 6.58T + 19T^{2} \)
23 \( 1 + 5.58T + 23T^{2} \)
29 \( 1 + 8.16T + 29T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 1.58T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 + 9.58T + 53T^{2} \)
59 \( 1 - 4.58T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 8.58T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 9.16T + 89T^{2} \)
97 \( 1 + 2.41T + 97T^{2} \)
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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

−12.47963995589029819295526310648, −11.50530040163972206601299999110, −10.36938564096962006674759496066, −9.577806794640621249754897573339, −8.161939317897230989109662638355, −6.53380845063870952667786544858, −5.75369936884517779737088399372, −5.11749517832103761916153067282, −3.75271478128334122181597471475, −2.06016902653797192392454285505, 2.06016902653797192392454285505, 3.75271478128334122181597471475, 5.11749517832103761916153067282, 5.75369936884517779737088399372, 6.53380845063870952667786544858, 8.161939317897230989109662638355, 9.577806794640621249754897573339, 10.36938564096962006674759496066, 11.50530040163972206601299999110, 12.47963995589029819295526310648

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