Properties

Label 2-231-33.32-c1-0-22
Degree $2$
Conductor $231$
Sign $0.768 + 0.640i$
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  + 2.45·2-s + (−1.72 − 0.157i)3-s + 4.01·4-s − 3.63i·5-s + (−4.22 − 0.385i)6-s i·7-s + 4.92·8-s + (2.95 + 0.542i)9-s − 8.91i·10-s + (−1.88 + 2.72i)11-s + (−6.91 − 0.630i)12-s + 5.76i·13-s − 2.45i·14-s + (−0.571 + 6.27i)15-s + 4.06·16-s + 4.19·17-s + ⋯
L(s)  = 1  + 1.73·2-s + (−0.995 − 0.0907i)3-s + 2.00·4-s − 1.62i·5-s + (−1.72 − 0.157i)6-s − 0.377i·7-s + 1.74·8-s + (0.983 + 0.180i)9-s − 2.81i·10-s + (−0.568 + 0.822i)11-s + (−1.99 − 0.181i)12-s + 1.59i·13-s − 0.655i·14-s + (−0.147 + 1.61i)15-s + 1.01·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.768 + 0.640i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.768 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18530 - 0.791566i\)
\(L(\frac12)\) \(\approx\) \(2.18530 - 0.791566i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.157i)T \)
7 \( 1 + iT \)
11 \( 1 + (1.88 - 2.72i)T \)
good2 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 + 3.63iT - 5T^{2} \)
13 \( 1 - 5.76iT - 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 - 2.05iT - 19T^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 - 4.50T + 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 0.628T + 37T^{2} \)
41 \( 1 - 1.39T + 41T^{2} \)
43 \( 1 + 0.462iT - 43T^{2} \)
47 \( 1 - 4.50iT - 47T^{2} \)
53 \( 1 + 1.38iT - 53T^{2} \)
59 \( 1 + 0.774iT - 59T^{2} \)
61 \( 1 - 5.90iT - 61T^{2} \)
67 \( 1 + 0.632T + 67T^{2} \)
71 \( 1 + 9.61iT - 71T^{2} \)
73 \( 1 + 14.3iT - 73T^{2} \)
79 \( 1 - 7.56iT - 79T^{2} \)
83 \( 1 - 6.21T + 83T^{2} \)
89 \( 1 + 5.60iT - 89T^{2} \)
97 \( 1 + 8.98T + 97T^{2} \)
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

−12.29017802612851438583927108220, −11.77421256670448954631770466377, −10.56150593361501225637743840132, −9.343169225200480751170165319958, −7.67833054532870463811723269328, −6.58704681180218379459053514404, −5.48723617420018659309219799393, −4.72900564122502442347292085976, −4.10254479858703210372091671226, −1.69939978010165784888397619548, 2.80439336392556313407222723438, 3.59441097309565515358261581276, 5.34952614481961043884746521745, 5.76745314509541733729376881033, 6.78756780024381109920599638650, 7.72340695720564725948494148622, 10.10533058846905708928942962962, 10.84254045764394072597471569749, 11.40473622877074011350893016931, 12.38388664389573494184093784776

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