Properties

Label 2-231-7.2-c1-0-7
Degree $2$
Conductor $231$
Sign $0.248 - 0.968i$
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  + (−1.17 + 2.03i)2-s + (0.5 + 0.866i)3-s + (−1.76 − 3.06i)4-s + (2.05 − 3.56i)5-s − 2.35·6-s + (2.51 + 0.818i)7-s + 3.60·8-s + (−0.499 + 0.866i)9-s + (4.84 + 8.38i)10-s + (−0.5 − 0.866i)11-s + (1.76 − 3.06i)12-s + 5.39·13-s + (−4.62 + 4.16i)14-s + 4.11·15-s + (−0.709 + 1.22i)16-s + (−1.17 − 2.03i)17-s + ⋯
L(s)  = 1  + (−0.831 + 1.44i)2-s + (0.288 + 0.499i)3-s + (−0.883 − 1.53i)4-s + (0.920 − 1.59i)5-s − 0.960·6-s + (0.950 + 0.309i)7-s + 1.27·8-s + (−0.166 + 0.288i)9-s + (1.53 + 2.65i)10-s + (−0.150 − 0.261i)11-s + (0.510 − 0.883i)12-s + 1.49·13-s + (−1.23 + 1.11i)14-s + 1.06·15-s + (−0.177 + 0.307i)16-s + (−0.285 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.248 - 0.968i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842088 + 0.653217i\)
\(L(\frac12)\) \(\approx\) \(0.842088 + 0.653217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.51 - 0.818i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.17 - 2.03i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.05 + 3.56i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 + (1.17 + 2.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.87 - 4.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.709 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.13T + 29T^{2} \)
31 \( 1 + (-0.266 - 0.461i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.68T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (4.47 - 7.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.80 + 3.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.709 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.88 - 3.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.01 + 1.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 + (3.23 + 5.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.09 - 8.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + (-4.36 + 7.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−12.64566834798314015571229904412, −11.16911189617654953543823003564, −9.918371179069197331855227924345, −9.039812651555232840614928780519, −8.540766858810856787226946560850, −7.87912836298992591809524252854, −6.04647409418166810248665668402, −5.51416607077757333680433547930, −4.41962153051432399058732815301, −1.47775343198064143655500908774, 1.68196289510142251257437016136, 2.57347526539056940725661369465, 3.84629497207179862484380209848, 6.05697915605608702771997866517, 7.18426598734566877054927351694, 8.346698489323293580839864126851, 9.258703826921222224870535114212, 10.41800467263464434925934293301, 10.94920130415052621608813626953, 11.44894033763919226702422280870

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