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Properties

Label 2-231-11.4-c1-0-6
Degree $2$
Conductor $231$
Sign $0.116 + 0.993i$
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 − 0.363i)2-s + (−0.309 + 0.951i)3-s + (−0.5 − 1.53i)4-s + (1.30 − 0.951i)5-s + (0.5 − 0.363i)6-s + (−0.309 − 0.951i)7-s + (−0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s − 10-s + (1.69 − 2.85i)11-s + 1.61·12-s + (−3.42 − 2.48i)13-s + (−0.190 + 0.587i)14-s + (0.499 + 1.53i)15-s + (−1.49 + 1.08i)16-s + (2.80 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.256i)2-s + (−0.178 + 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.585 − 0.425i)5-s + (0.204 − 0.148i)6-s + (−0.116 − 0.359i)7-s + (−0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s − 0.316·10-s + (0.509 − 0.860i)11-s + 0.467·12-s + (−0.950 − 0.690i)13-s + (−0.0510 + 0.157i)14-s + (0.129 + 0.397i)15-s + (−0.374 + 0.272i)16-s + (0.681 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685093 - 0.609235i\)
\(L(\frac12)\) \(\approx\) \(0.685093 - 0.609235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-1.69 + 2.85i)T \)
good2 \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.42 + 2.48i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.80 + 2.04i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2 + 6.15i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 + (0.0729 + 0.224i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.42 + 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.85 - 5.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.04 - 9.37i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (1.59 - 4.89i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.1 - 7.38i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.42 + 4.39i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.42 + 4.66i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + (-3.66 + 2.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.28 - 7.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.92 - 3.57i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.85 - 4.97i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 + (-4.28 - 3.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.59345535780682501524068540869, −10.95993559539669512538554486855, −9.727981012917189621149053394049, −9.531046756295636685489740233683, −8.369403849185161650183502390131, −6.79245754156162379511386081365, −5.46051370661874869784941089377, −4.89309863858823884918400045961, −3.00778686719332393471852774768, −0.921556623080204601596263201403, 2.09350870517998568886041453917, 3.69132467307026662966969430440, 5.32247828989768773616474982148, 6.68999949061032228405522961418, 7.28940874036668505933297838907, 8.428063946224317755795563440833, 9.486606065151950118578990536405, 10.22657074812975423694318445227, 11.88520573551656959164564504727, 12.26666157060904168190158223651

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