Properties

Label 2-231-11.3-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.999 + 0.0439i$
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  + (−2.11 + 1.53i)2-s + (−0.309 − 0.951i)3-s + (1.5 − 4.61i)4-s + (−1.30 − 0.951i)5-s + (2.11 + 1.53i)6-s + (0.309 − 0.951i)7-s + (2.30 + 7.10i)8-s + (−0.809 + 0.587i)9-s + 4.23·10-s + (−2.54 + 2.12i)11-s − 4.85·12-s + (−4.42 + 3.21i)13-s + (0.809 + 2.48i)14-s + (−0.499 + 1.53i)15-s + (−7.97 − 5.79i)16-s + (2.42 + 1.76i)17-s + ⋯
L(s)  = 1  + (−1.49 + 1.08i)2-s + (−0.178 − 0.549i)3-s + (0.750 − 2.30i)4-s + (−0.585 − 0.425i)5-s + (0.864 + 0.628i)6-s + (0.116 − 0.359i)7-s + (0.816 + 2.51i)8-s + (−0.269 + 0.195i)9-s + 1.33·10-s + (−0.767 + 0.641i)11-s − 1.40·12-s + (−1.22 + 0.892i)13-s + (0.216 + 0.665i)14-s + (−0.129 + 0.397i)15-s + (−1.99 − 1.44i)16-s + (0.588 + 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.999 + 0.0439i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.999 + 0.0439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00202661 - 0.0921638i\)
\(L(\frac12)\) \(\approx\) \(0.00202661 - 0.0921638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (2.54 - 2.12i)T \)
good2 \( 1 + (2.11 - 1.53i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.30 + 0.951i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.42 - 3.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.42 - 1.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + (-0.927 + 2.85i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.190 - 0.138i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.61 - 8.05i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.954 + 2.93i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-3.11 - 9.59i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.30 - 2.40i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.89 + 12.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.42 - 3.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 + (9.89 + 7.19i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.19 - 3.66i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.54 - 4.02i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.38 + 1.73i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.67T + 89T^{2} \)
97 \( 1 + (-13.2 + 9.64i)T + (29.9 - 92.2i)T^{2} \)
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

−12.40359623213200877222693095870, −11.67285586420006387315683342130, −10.21096154541276022104112759940, −9.804584641946643686139246666044, −8.331843907894732227599571819918, −7.82987854008509418057086213217, −7.06914598955019881708500646147, −5.95177059252378348657252491087, −4.67417751463471007293925104374, −1.80033546662995535019263700637, 0.11886784829226630252888362140, 2.56220895859971984361298126914, 3.50248675518077223148018888677, 5.31142028221977860744088455203, 7.27785477966797319823951118705, 7.993898932392597326638770060303, 8.973432883554362188689702707892, 10.04772605988212059908424381759, 10.50405368132859352427832258771, 11.57204646252305713031651721673

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