Properties

Label 2-231-231.230-c0-0-2
Degree $2$
Conductor $231$
Sign $1$
Analytic cond. $0.115284$
Root an. cond. $0.339535$
Motivic weight $0$
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 + 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 13-s + 14-s − 15-s − 16-s + 18-s − 19-s − 21-s − 22-s + 24-s − 26-s − 27-s + 29-s − 30-s + 33-s + 35-s − 37-s − 38-s + 39-s + ⋯
L(s)  = 1  + 2-s − 3-s + 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 13-s + 14-s − 15-s − 16-s + 18-s − 19-s − 21-s − 22-s + 24-s − 26-s − 27-s + 29-s − 30-s + 33-s + 35-s − 37-s − 38-s + 39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.115284\)
Root analytic conductor: \(0.339535\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{231} (230, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9001109165\)
\(L(\frac12)\) \(\approx\) \(0.9001109165\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
5 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.52774274759651952795141339305, −11.73652704061383165115573225475, −10.60078208166062937669383107989, −9.869212863884393600082192228144, −8.497126690422111619558314566805, −7.05474688251435185853047927074, −5.82109089115999547293789376839, −5.18815546221459090870871092600, −4.38576489093358455968417944071, −2.30109140277429904126234426023, 2.30109140277429904126234426023, 4.38576489093358455968417944071, 5.18815546221459090870871092600, 5.82109089115999547293789376839, 7.05474688251435185853047927074, 8.497126690422111619558314566805, 9.869212863884393600082192228144, 10.60078208166062937669383107989, 11.73652704061383165115573225475, 12.52774274759651952795141339305

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