↑

Properties

Label 2-231-33.32-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.959 + 0.282i$
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  − 1.36·2-s + (−0.859 + 1.50i)3-s − 0.132·4-s + 3.05i·5-s + (1.17 − 2.05i)6-s + i·7-s + 2.91·8-s + (−1.52 − 2.58i)9-s − 4.17i·10-s + (−2.29 + 2.39i)11-s + (0.113 − 0.198i)12-s − 3.62i·13-s − 1.36i·14-s + (−4.59 − 2.62i)15-s − 3.71·16-s − 4.38·17-s + ⋯
L(s)  = 1  − 0.966·2-s + (−0.496 + 0.868i)3-s − 0.0661·4-s + 1.36i·5-s + (0.479 − 0.838i)6-s + 0.377i·7-s + 1.03·8-s + (−0.507 − 0.861i)9-s − 1.32i·10-s + (−0.692 + 0.721i)11-s + (0.0328 − 0.0574i)12-s − 1.00i·13-s − 0.365i·14-s + (−1.18 − 0.678i)15-s − 0.929·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.959 + 0.282i$
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: \((2,\ 231,\ (\ :1/2),\ -0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0411308 - 0.285616i\)
\(L(\frac12)\) \(\approx\) \(0.0411308 - 0.285616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.859 - 1.50i)T \)
7 \( 1 - iT \)
11 \( 1 + (2.29 - 2.39i)T \)
good2 \( 1 + 1.36T + 2T^{2} \)
5 \( 1 - 3.05iT - 5T^{2} \)
13 \( 1 + 3.62iT - 13T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 - 1.58iT - 19T^{2} \)
23 \( 1 + 2.62iT - 23T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 + 7.41T + 37T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 - 5.36iT - 43T^{2} \)
47 \( 1 + 4.01iT - 47T^{2} \)
53 \( 1 - 6.53iT - 53T^{2} \)
59 \( 1 - 9.19iT - 59T^{2} \)
61 \( 1 - 7.21iT - 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 + 1.54iT - 71T^{2} \)
73 \( 1 + 7.32iT - 73T^{2} \)
79 \( 1 - 16.5iT - 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 - 6.22iT - 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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

−12.53394540635765923052460218717, −11.19430995988573333216437392337, −10.55797904106536554489003176895, −10.03517492271853051448108749975, −9.047513227768669431979417781521, −7.905704568047099412682724105489, −6.83396035677366190512355401059, −5.55287582562371775477214218947, −4.25712306676798751835429703021, −2.68770881844478074229119822339, 0.34490146077986751923827388280, 1.75061402159665175498096464850, 4.42861425950657997918600048723, 5.41369126732654034361733235941, 6.87184818167840575766783076017, 7.929016193477228048607291060460, 8.677038966705774504324239576157, 9.435939428925381869208366851488, 10.79213945305477098308102838728, 11.52065885765591875921789452053

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