Properties

Label 2-231-11.9-c1-0-9
Degree $2$
Conductor $231$
Sign $-0.605 + 0.795i$
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  + (0.5 − 1.53i)2-s + (−0.809 + 0.587i)3-s + (−0.5 − 0.363i)4-s + (−0.809 − 2.48i)5-s + (0.5 + 1.53i)6-s + (−0.809 − 0.587i)7-s + (1.80 − 1.31i)8-s + (0.309 − 0.951i)9-s − 4.23·10-s + (−0.809 − 3.21i)11-s + 0.618·12-s + (−0.309 + 0.951i)13-s + (−1.30 + 0.951i)14-s + (2.11 + 1.53i)15-s + (−1.50 − 4.61i)16-s + (0.0729 + 0.224i)17-s + ⋯
L(s)  = 1  + (0.353 − 1.08i)2-s + (−0.467 + 0.339i)3-s + (−0.250 − 0.181i)4-s + (−0.361 − 1.11i)5-s + (0.204 + 0.628i)6-s + (−0.305 − 0.222i)7-s + (0.639 − 0.464i)8-s + (0.103 − 0.317i)9-s − 1.33·10-s + (−0.243 − 0.969i)11-s + 0.178·12-s + (−0.0857 + 0.263i)13-s + (−0.349 + 0.254i)14-s + (0.546 + 0.397i)15-s + (−0.375 − 1.15i)16-s + (0.0176 + 0.0544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.605 + 0.795i$
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: \((2,\ 231,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539042 - 1.08731i\)
\(L(\frac12)\) \(\approx\) \(0.539042 - 1.08731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 3.21i)T \)
good2 \( 1 + (-0.5 + 1.53i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.0729 - 0.224i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
29 \( 1 + (-5.42 - 3.94i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.78 - 5.48i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.85 - 6.43i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.92 + 3.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (6.73 - 4.89i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.66 + 5.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.78 - 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.16 + 9.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.76T + 67T^{2} \)
71 \( 1 + (-0.454 - 1.40i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.92 - 5.03i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.28 + 3.94i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.85 + 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.61T + 89T^{2} \)
97 \( 1 + (5.25 - 16.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.84625423488852839552549668308, −11.08229570766009050674271406798, −10.23744283100220446027637630632, −9.163233764456512863131814112677, −8.093732273481845721941322941203, −6.64677069365387458156733330604, −5.16379623109223910251492358918, −4.25965419056545079854625141487, −3.08692683359994346296652748293, −1.01902304890237170695972190348, 2.49997133089484997108409607897, 4.35985454269716408791137110575, 5.64229514701566240264565907388, 6.55263229769220659941255111949, 7.27924133666319223008702019418, 8.017949150503181425676963029325, 9.740898070325001133670321385741, 10.74784541665669060647311003486, 11.53232134773781330176614602380, 12.68417816933736135662617503708

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