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Properties

Label 2-231-7.2-c1-0-1
Degree $2$
Conductor $231$
Sign $-0.988 + 0.151i$
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  + (−0.643 + 1.11i)2-s + (−0.5 − 0.866i)3-s + (0.171 + 0.296i)4-s + (−1.95 + 3.39i)5-s + 1.28·6-s + (−0.234 − 2.63i)7-s − 3.01·8-s + (−0.499 + 0.866i)9-s + (−2.52 − 4.36i)10-s + (0.5 + 0.866i)11-s + (0.171 − 0.296i)12-s − 3.04·13-s + (3.08 + 1.43i)14-s + 3.91·15-s + (1.59 − 2.76i)16-s + (−1.98 − 3.44i)17-s + ⋯
L(s)  = 1  + (−0.455 + 0.788i)2-s + (−0.288 − 0.499i)3-s + (0.0856 + 0.148i)4-s + (−0.875 + 1.51i)5-s + 0.525·6-s + (−0.0885 − 0.996i)7-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (−0.797 − 1.38i)10-s + (0.150 + 0.261i)11-s + (0.0494 − 0.0856i)12-s − 0.843·13-s + (0.825 + 0.383i)14-s + 1.01·15-s + (0.399 − 0.692i)16-s + (−0.481 − 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.988 + 0.151i$
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: \((2,\ 231,\ (\ :1/2),\ -0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0307296 - 0.403537i\)
\(L(\frac12)\) \(\approx\) \(0.0307296 - 0.403537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.234 + 2.63i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.643 - 1.11i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.95 - 3.39i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.04T + 13T^{2} \)
17 \( 1 + (1.98 + 3.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.79 - 6.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.25 - 3.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.75T + 29T^{2} \)
31 \( 1 + (-3.37 - 5.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.171 - 0.296i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.79T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (0.828 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.47 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.83 - 3.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.234 + 0.405i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.28 - 2.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 + (-4.36 - 7.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.359 - 0.623i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (-6.17 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−12.40795757076901436179449544623, −11.77763405584022896184656004037, −10.77522616575810057668168663455, −9.921774787860672459287891780376, −8.281146135391031738933654017527, −7.34952197952317701573599446982, −7.09198929617339378087978529030, −6.12406447630171194011383237585, −4.09395285296385722055944087953, −2.80079855704437018519362357019, 0.37462168265862799602655610699, 2.39691101137453934269770924319, 4.21052771748260298562045961778, 5.19687492371404517808090140627, 6.39266895174126317941057705265, 8.337783744399611859696992072970, 8.877639392898293611578766445871, 9.677783754543208706215787909273, 10.86847088658745242933774646122, 11.72294593244632598817469501305

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