↑

Properties

Label 2-231-11.4-c1-0-11
Degree $2$
Conductor $231$
Sign $-0.996 - 0.0784i$
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  + (−0.383 − 0.278i)2-s + (0.309 − 0.951i)3-s + (−0.548 − 1.68i)4-s + (−3.04 + 2.20i)5-s + (−0.383 + 0.278i)6-s + (−0.309 − 0.951i)7-s + (−0.552 + 1.70i)8-s + (−0.809 − 0.587i)9-s + 1.78·10-s + (−2.47 − 2.21i)11-s − 1.77·12-s + (−2.19 − 1.59i)13-s + (−0.146 + 0.450i)14-s + (1.16 + 3.57i)15-s + (−2.18 + 1.58i)16-s + (−2.00 + 1.45i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.196i)2-s + (0.178 − 0.549i)3-s + (−0.274 − 0.844i)4-s + (−1.35 + 0.988i)5-s + (−0.156 + 0.113i)6-s + (−0.116 − 0.359i)7-s + (−0.195 + 0.601i)8-s + (−0.269 − 0.195i)9-s + 0.563·10-s + (−0.744 − 0.667i)11-s − 0.512·12-s + (−0.607 − 0.441i)13-s + (−0.0391 + 0.120i)14-s + (0.299 + 0.923i)15-s + (−0.546 + 0.397i)16-s + (−0.487 + 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.996 - 0.0784i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.996 - 0.0784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0123513 + 0.314458i\)
\(L(\frac12)\) \(\approx\) \(0.0123513 + 0.314458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.47 + 2.21i)T \)
good2 \( 1 + (0.383 + 0.278i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (3.04 - 2.20i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.19 + 1.59i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.00 - 1.45i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.539 - 1.65i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 + (1.79 + 5.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.81 + 1.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.36 + 7.27i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.29 - 3.98i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 + (-3.83 + 11.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.15 + 5.19i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.40 - 4.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.28 + 6.74i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (6.27 - 4.56i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.84 - 14.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.70 - 4.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.62 - 6.99i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 + (5.66 + 4.11i)T + (29.9 + 92.2i)T^{2} \)
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

−11.37041157279239631305890894074, −10.89377361142575387407398711622, −9.985413455941138331121238138327, −8.584242260720897353655285912044, −7.72955690453405386425814010235, −6.82984701557832788363332163346, −5.56285610047923336698732166974, −3.96754653285426928898764738516, −2.59047508505105749147923719062, −0.26632730924868451666558637247, 3.01861004936544629385048454703, 4.35905469210680976435668889982, 4.94973057793470982640242341266, 7.13390880982550371666748681431, 7.83990983785731387795228425580, 8.876157099437686194056712065773, 9.298650817444329769896193984385, 10.84926613591557479303593038275, 11.95767267803105221818413073065, 12.50963819986559320272128112920

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