↑

Properties

Label 2-231-11.9-c1-0-6
Degree $2$
Conductor $231$
Sign $0.569 + 0.821i$
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  + (0.449 − 1.38i)2-s + (0.809 − 0.587i)3-s + (−0.0967 − 0.0703i)4-s + (0.737 + 2.27i)5-s + (−0.449 − 1.38i)6-s + (0.809 + 0.587i)7-s + (2.21 − 1.60i)8-s + (0.309 − 0.951i)9-s + 3.47·10-s + (−3.30 − 0.224i)11-s − 0.119·12-s + (−0.281 + 0.867i)13-s + (1.17 − 0.855i)14-s + (1.93 + 1.40i)15-s + (−1.30 − 4.01i)16-s + (−0.675 − 2.07i)17-s + ⋯
L(s)  = 1  + (0.318 − 0.979i)2-s + (0.467 − 0.339i)3-s + (−0.0483 − 0.0351i)4-s + (0.329 + 1.01i)5-s + (−0.183 − 0.565i)6-s + (0.305 + 0.222i)7-s + (0.783 − 0.568i)8-s + (0.103 − 0.317i)9-s + 1.09·10-s + (−0.997 − 0.0676i)11-s − 0.0345·12-s + (−0.0781 + 0.240i)13-s + (0.314 − 0.228i)14-s + (0.498 + 0.362i)15-s + (−0.326 − 1.00i)16-s + (−0.163 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63014 - 0.853299i\)
\(L(\frac12)\) \(\approx\) \(1.63014 - 0.853299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.30 + 0.224i)T \)
good2 \( 1 + (-0.449 + 1.38i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.737 - 2.27i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.281 - 0.867i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.675 + 2.07i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.575 - 0.418i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.80T + 23T^{2} \)
29 \( 1 + (7.17 + 5.21i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.04 - 6.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.40 - 2.47i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.67 + 7.02i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.51T + 43T^{2} \)
47 \( 1 + (5.07 - 3.68i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.02 - 6.24i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.78 - 5.65i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.0193 + 0.0596i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (1.84 + 5.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.54 - 2.57i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.50 - 4.61i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.10 - 9.55i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.261T + 89T^{2} \)
97 \( 1 + (-3.05 + 9.40i)T + (-78.4 - 57.0i)T^{2} \)
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

−12.04379358814174241516588059545, −11.08286120541542916712983014787, −10.41384779422841808661843905059, −9.434848281434137180755968569989, −7.921551426128284886871589003040, −7.18133027523369205282680099048, −5.87886460490473163317438962746, −4.18187573610993624659485660980, −2.84608626973247902184466851525, −2.10385443391859311097296875415, 2.03895737164432998325077806206, 4.17908474513799843047751550288, 5.19040582460461893143084517402, 5.99146375651087857172067772424, 7.57840133793551966974797299162, 8.111026646096794219317264861712, 9.250232080370887109149590744811, 10.33373439290330746284296586586, 11.26584884373353845130340240292, 12.86411333348462029158789935206

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