Properties

Label 2-231-1.1-c1-0-1
Degree $2$
Conductor $231$
Sign $1$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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  − 2-s − 3-s − 4-s − 2·5-s + 6-s + 7-s + 3·8-s + 9-s + 2·10-s − 11-s + 12-s + 6·13-s − 14-s + 2·15-s − 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s − 21-s + 22-s − 3·24-s − 25-s − 6·26-s − 27-s − 28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s + 0.213·22-s − 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-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: yes
Arithmetic: yes
Character: $\chi_{231} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5822888619\)
\(L(\frac12)\) \(\approx\) \(0.5822888619\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−11.89946826402131998655781888440, −11.18319190967266805266750560108, −10.32896954095524943775333918549, −9.238375783455941661417111092919, −8.146329212122427589249395398868, −7.61386400817600432579143653551, −6.06507120539099918049903981653, −4.78033654342364343647378297963, −3.69478633591386645327522922963, −1.02008449562817059954899874886, 1.02008449562817059954899874886, 3.69478633591386645327522922963, 4.78033654342364343647378297963, 6.06507120539099918049903981653, 7.61386400817600432579143653551, 8.146329212122427589249395398868, 9.238375783455941661417111092919, 10.32896954095524943775333918549, 11.18319190967266805266750560108, 11.89946826402131998655781888440

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