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Properties

Label 2-231-77.54-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.366 - 0.930i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 − 1.29i)2-s + (−0.866 + 0.5i)3-s + (2.34 + 4.06i)4-s + (−1.30 − 0.751i)5-s + 2.58·6-s + (1.55 − 2.13i)7-s − 6.95i·8-s + (0.499 − 0.866i)9-s + (1.94 + 3.36i)10-s + (−2.72 + 1.89i)11-s + (−4.06 − 2.34i)12-s − 3.99·13-s + (−6.25 + 2.77i)14-s + 1.50·15-s + (−4.30 + 7.46i)16-s + (1.38 + 2.39i)17-s + ⋯
L(s)  = 1  + (−1.58 − 0.914i)2-s + (−0.499 + 0.288i)3-s + (1.17 + 2.03i)4-s + (−0.581 − 0.335i)5-s + 1.05·6-s + (0.588 − 0.808i)7-s − 2.45i·8-s + (0.166 − 0.288i)9-s + (0.614 + 1.06i)10-s + (−0.821 + 0.570i)11-s + (−1.17 − 0.676i)12-s − 1.10·13-s + (−1.67 + 0.742i)14-s + 0.387·15-s + (−1.07 + 1.86i)16-s + (0.335 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.366 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0384208 + 0.0564094i\)
\(L(\frac12)\) \(\approx\) \(0.0384208 + 0.0564094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-1.55 + 2.13i)T \)
11 \( 1 + (2.72 - 1.89i)T \)
good2 \( 1 + (2.23 + 1.29i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.30 + 0.751i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.99T + 13T^{2} \)
17 \( 1 + (-1.38 - 2.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.45 - 4.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0359 + 0.0623i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.06iT - 29T^{2} \)
31 \( 1 + (8.84 - 5.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.90 - 6.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.98T + 41T^{2} \)
43 \( 1 + 5.41iT - 43T^{2} \)
47 \( 1 + (-4.48 - 2.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.23 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.52 - 3.18i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.688 + 1.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.27 - 3.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.59T + 71T^{2} \)
73 \( 1 + (4.48 + 7.76i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.0 - 5.82i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.34T + 83T^{2} \)
89 \( 1 + (11.7 + 6.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.63iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.28186830819654753471647977123, −11.22023054248338607240139016659, −10.38619316060711124712599591119, −9.997282882094721008247800843571, −8.629244879350503726120327237275, −7.81612606489089829401996172063, −7.06540085506984962873859713158, −4.95938560954454948868556666881, −3.62900166071352907956148636805, −1.76439681195150665651879287051, 0.094301069550416512058691688601, 2.30657253161051844049297817439, 5.10758146758620732509999706745, 5.99225458892304753905854165798, 7.34378217739307142709487469841, 7.71201214965903443414904286384, 8.824578352420549952570729360453, 9.727926423653274862171833929436, 10.96683471200590157145662915797, 11.36128547943516480909227096210

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