↑

Properties

Label 2-231-231.65-c1-0-6
Degree $2$
Conductor $231$
Sign $0.998 + 0.0476i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.463 − 0.803i)2-s + (−1.71 + 0.254i)3-s + (0.569 + 0.986i)4-s + (−1.53 − 0.886i)5-s + (−0.590 + 1.49i)6-s + (2.26 + 1.36i)7-s + 2.91·8-s + (2.87 − 0.871i)9-s + (−1.42 + 0.822i)10-s + (3.22 + 0.791i)11-s + (−1.22 − 1.54i)12-s + 0.151i·13-s + (2.14 − 1.19i)14-s + (2.85 + 1.12i)15-s + (0.212 − 0.368i)16-s + (0.598 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.328 − 0.568i)2-s + (−0.989 + 0.146i)3-s + (0.284 + 0.493i)4-s + (−0.686 − 0.396i)5-s + (−0.241 + 0.610i)6-s + (0.857 + 0.514i)7-s + 1.02·8-s + (0.956 − 0.290i)9-s + (−0.450 + 0.260i)10-s + (0.971 + 0.238i)11-s + (−0.354 − 0.445i)12-s + 0.0421i·13-s + (0.573 − 0.318i)14-s + (0.737 + 0.291i)15-s + (0.0531 − 0.0920i)16-s + (0.145 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.998 + 0.0476i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.998 + 0.0476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23960 - 0.0295286i\)
\(L(\frac12)\) \(\approx\) \(1.23960 - 0.0295286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.254i)T \)
7 \( 1 + (-2.26 - 1.36i)T \)
11 \( 1 + (-3.22 - 0.791i)T \)
good2 \( 1 + (-0.463 + 0.803i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.53 + 0.886i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.151iT - 13T^{2} \)
17 \( 1 + (-0.598 - 1.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.13 - 2.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.52 + 0.880i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.83T + 29T^{2} \)
31 \( 1 + (2.85 + 4.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + (1.28 + 0.739i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.94 - 2.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.850 + 0.490i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.24 + 3.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.75 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.53iT - 71T^{2} \)
73 \( 1 + (9.61 - 5.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.02 - 4.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.653T + 83T^{2} \)
89 \( 1 + (0.343 + 0.198i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.93T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03173281016893206916585671018, −11.54433626675421598910701045304, −10.72907355350161052658965453131, −9.429782572160612698243671991554, −8.088487133997472648832826370580, −7.24659561318605294286842483444, −5.82064125190087789287246339704, −4.58201162270584527629083631736, −3.77274721024053155164036281238, −1.66547957182242196055385475590, 1.33966239110890023149246893809, 3.97865005607553723228108516107, 5.06307736347423619940873748734, 6.04247341093139563533885482758, 7.23239484628425385793965409363, 7.56588582319881001504328210522, 9.421038239969165367846831038181, 10.71651718201995296344020829010, 11.24513483180284327094043257214, 11.87824033743511256268319326087

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