Properties

Label 2-231-77.54-c1-0-11
Degree $2$
Conductor $231$
Sign $0.612 - 0.790i$
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  + (2.16 + 1.25i)2-s + (0.866 − 0.5i)3-s + (2.13 + 3.69i)4-s + (−0.849 − 0.490i)5-s + 2.50·6-s + (−1.62 − 2.08i)7-s + 5.66i·8-s + (0.499 − 0.866i)9-s + (−1.22 − 2.12i)10-s + (−0.488 + 3.28i)11-s + (3.69 + 2.13i)12-s − 0.941·13-s + (−0.916 − 6.55i)14-s − 0.980·15-s + (−2.83 + 4.90i)16-s + (−2.58 − 4.48i)17-s + ⋯
L(s)  = 1  + (1.53 + 0.884i)2-s + (0.499 − 0.288i)3-s + (1.06 + 1.84i)4-s + (−0.379 − 0.219i)5-s + 1.02·6-s + (−0.615 − 0.788i)7-s + 2.00i·8-s + (0.166 − 0.288i)9-s + (−0.388 − 0.672i)10-s + (−0.147 + 0.989i)11-s + (1.06 + 0.615i)12-s − 0.261·13-s + (−0.244 − 1.75i)14-s − 0.253·15-s + (−0.707 + 1.22i)16-s + (−0.627 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44483 + 1.19818i\)
\(L(\frac12)\) \(\approx\) \(2.44483 + 1.19818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (1.62 + 2.08i)T \)
11 \( 1 + (0.488 - 3.28i)T \)
good2 \( 1 + (-2.16 - 1.25i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.849 + 0.490i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.941T + 13T^{2} \)
17 \( 1 + (2.58 + 4.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.27 - 2.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.379 + 0.656i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.74iT - 29T^{2} \)
31 \( 1 + (-5.50 + 3.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.49 - 2.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.80T + 41T^{2} \)
43 \( 1 + 6.73iT - 43T^{2} \)
47 \( 1 + (-8.42 - 4.86i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.24 - 3.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.7 - 7.35i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.81 - 4.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.21 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (5.14 + 8.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.63 - 3.82i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.28T + 83T^{2} \)
89 \( 1 + (-2.06 - 1.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.30iT - 97T^{2} \)
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

−12.45012751156841329555732649513, −12.14485041846112275926739558455, −10.55179655098704185993415771674, −9.221973893807696012015961460751, −7.77633804084522273093557347732, −7.18470432903997725634115588288, −6.31971498644073429353941115917, −4.80629270295986375181103971401, −4.04156683053964538862167982232, −2.75045246288645286825012986369, 2.35204096765910839870102445910, 3.31631408952728862320762583463, 4.29271818932145482372196944156, 5.61547660035094253061774657351, 6.48749647849695029627686746979, 8.204299069633089515484784806492, 9.389841910056547392798997033442, 10.53103949448740655318294601184, 11.30636481613144031190899693127, 12.15870677893176268926845023229

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