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Properties

Label 2-231-7.2-c1-0-3
Degree $2$
Conductor $231$
Sign $-0.160 - 0.987i$
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  + (−1.39 + 2.40i)2-s + (−0.5 − 0.866i)3-s + (−2.86 − 4.96i)4-s + (−0.412 + 0.715i)5-s + 2.78·6-s + (2.63 − 0.257i)7-s + 10.3·8-s + (−0.499 + 0.866i)9-s + (−1.14 − 1.98i)10-s + (0.5 + 0.866i)11-s + (−2.86 + 4.96i)12-s − 0.296·13-s + (−3.04 + 6.70i)14-s + 0.825·15-s + (−8.72 + 15.1i)16-s + (3.34 + 5.79i)17-s + ⋯
L(s)  = 1  + (−0.983 + 1.70i)2-s + (−0.288 − 0.499i)3-s + (−1.43 − 2.48i)4-s + (−0.184 + 0.319i)5-s + 1.13·6-s + (0.995 − 0.0971i)7-s + 3.67·8-s + (−0.166 + 0.288i)9-s + (−0.363 − 0.629i)10-s + (0.150 + 0.261i)11-s + (−0.828 + 1.43i)12-s − 0.0823·13-s + (−0.813 + 1.79i)14-s + 0.213·15-s + (−2.18 + 3.77i)16-s + (0.811 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.160 - 0.987i$
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: \((2,\ 231,\ (\ :1/2),\ -0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445395 + 0.523402i\)
\(L(\frac12)\) \(\approx\) \(0.445395 + 0.523402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.257i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.39 - 2.40i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.412 - 0.715i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 + (-3.34 - 5.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.98 + 3.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.484T + 29T^{2} \)
31 \( 1 + (-3.66 - 6.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.86 + 4.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.645T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 + (3.86 - 6.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.55 + 6.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.578 - 1.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.63 - 4.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.50 + 2.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 + (8.01 + 13.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.16 + 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.37T + 83T^{2} \)
89 \( 1 + (6.08 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−12.62531066176668062747804583238, −11.07278821587219127500822308392, −10.42256420652522051984083745854, −9.149112959766927191402995598404, −8.202438772416156572894331406412, −7.51925726233525389224036345384, −6.66298002830413542276300542794, −5.62583662310436766154358543851, −4.59895188212412298440741821523, −1.33452186319518123630249285577, 1.05184918873545224089879531193, 2.78971114026863799858056733079, 4.13062593095988891962758771799, 5.13414987572860930729847082385, 7.55708143579859035272896148622, 8.342802264583581183374375800703, 9.348461615269938286962487492420, 10.02945340423348937771955248873, 11.09397033937807838378265122687, 11.68267300445339053664483930586

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