↑

Properties

Label 2-231-33.32-c1-0-21
Degree $2$
Conductor $231$
Sign $0.916 + 0.400i$
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  + 1.92·2-s + (1.23 − 1.21i)3-s + 1.69·4-s + 0.413i·5-s + (2.37 − 2.33i)6-s + i·7-s − 0.588·8-s + (0.0504 − 2.99i)9-s + 0.794i·10-s + (−3.07 + 1.23i)11-s + (2.09 − 2.05i)12-s + 1.35i·13-s + 1.92i·14-s + (0.502 + 0.510i)15-s − 4.51·16-s + 1.64·17-s + ⋯
L(s)  = 1  + 1.35·2-s + (0.713 − 0.701i)3-s + 0.846·4-s + 0.184i·5-s + (0.969 − 0.952i)6-s + 0.377i·7-s − 0.208·8-s + (0.0168 − 0.999i)9-s + 0.251i·10-s + (−0.927 + 0.372i)11-s + (0.603 − 0.593i)12-s + 0.376i·13-s + 0.513i·14-s + (0.129 + 0.131i)15-s − 1.12·16-s + 0.397·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.916 + 0.400i$
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: \((2,\ 231,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56640 - 0.535772i\)
\(L(\frac12)\) \(\approx\) \(2.56640 - 0.535772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.23 + 1.21i)T \)
7 \( 1 - iT \)
11 \( 1 + (3.07 - 1.23i)T \)
good2 \( 1 - 1.92T + 2T^{2} \)
5 \( 1 - 0.413iT - 5T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 - 4.21iT - 19T^{2} \)
23 \( 1 + 6.16iT - 23T^{2} \)
29 \( 1 + 2.73T + 29T^{2} \)
31 \( 1 - 4.56T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + 1.20iT - 43T^{2} \)
47 \( 1 - 8.33iT - 47T^{2} \)
53 \( 1 + 5.91iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 + 0.290iT - 73T^{2} \)
79 \( 1 + 16.6iT - 79T^{2} \)
83 \( 1 + 3.53T + 83T^{2} \)
89 \( 1 + 9.17iT - 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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

−12.57730585844089227337868751437, −11.70427221002173508095116556499, −10.32991129812269220660100858323, −9.051306139580426671372283800919, −8.030111583156851301249910321301, −6.86204192469498494303338064543, −5.91741726671534025297209662185, −4.67863257016366246328267924286, −3.33801139449036671140394849839, −2.30186153255410132985153768713, 2.73535453102813971319215462418, 3.66207180119843544939634549575, 4.85498447190192631819691313952, 5.54556784778045967235192390849, 7.15668621051490428754020589546, 8.331835626450380464121315734675, 9.375069291402953938694035930207, 10.49559776932026896072236061525, 11.37499702665927181758208166765, 12.62838087242916621262236137260

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