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Properties

Label 2-231-231.65-c1-0-26
Degree $2$
Conductor $231$
Sign $-0.883 + 0.467i$
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  + (1.19 − 2.07i)2-s + (−0.0197 − 1.73i)3-s + (−1.87 − 3.24i)4-s + (0.707 + 0.408i)5-s + (−3.61 − 2.03i)6-s + (2.64 − 0.0267i)7-s − 4.19·8-s + (−2.99 + 0.0683i)9-s + (1.69 − 0.979i)10-s + (−0.403 + 3.29i)11-s + (−5.58 + 3.30i)12-s + 5.85i·13-s + (3.11 − 5.52i)14-s + (0.693 − 1.23i)15-s + (−1.27 + 2.20i)16-s + (−1.19 − 2.06i)17-s + ⋯
L(s)  = 1  + (0.847 − 1.46i)2-s + (−0.0113 − 0.999i)3-s + (−0.936 − 1.62i)4-s + (0.316 + 0.182i)5-s + (−1.47 − 0.830i)6-s + (0.999 − 0.0100i)7-s − 1.48·8-s + (−0.999 + 0.0227i)9-s + (0.536 − 0.309i)10-s + (−0.121 + 0.992i)11-s + (−1.61 + 0.955i)12-s + 1.62i·13-s + (0.832 − 1.47i)14-s + (0.179 − 0.318i)15-s + (−0.318 + 0.552i)16-s + (−0.289 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.883 + 0.467i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461466 - 1.85900i\)
\(L(\frac12)\) \(\approx\) \(0.461466 - 1.85900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0197 + 1.73i)T \)
7 \( 1 + (-2.64 + 0.0267i)T \)
11 \( 1 + (0.403 - 3.29i)T \)
good2 \( 1 + (-1.19 + 2.07i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.707 - 0.408i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.85iT - 13T^{2} \)
17 \( 1 + (1.19 + 2.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.56 - 1.48i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.60 + 2.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.80T + 29T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.94 + 5.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.98T + 41T^{2} \)
43 \( 1 + 0.425iT - 43T^{2} \)
47 \( 1 + (9.65 + 5.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.33 - 0.769i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.11 + 0.644i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.56 + 2.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.729 - 1.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.85iT - 71T^{2} \)
73 \( 1 + (-2.07 + 1.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.34 + 2.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.53T + 83T^{2} \)
89 \( 1 + (-6.27 - 3.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.90T + 97T^{2} \)
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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

−11.86119132588652988510985131645, −11.31532660162800659011773490440, −10.19662848931450125569416305454, −9.142076998252069556361771716487, −7.74741833318156638799352301840, −6.55071927595366882593384760925, −5.17774464241189013469518812531, −4.17068974009890564322031082133, −2.37034701813849196914586886905, −1.67253743495720887886241660035, 3.31721535496648433121512189896, 4.52177624595985567010762747617, 5.53434652915567542090004571349, 5.94973004068644444087420771068, 7.86024884065519350726325832436, 8.195633796354934084989957768936, 9.483410274671126501231467173209, 10.74094212020302464997938161072, 11.64809145020942841917969533941, 13.13509782087453648656643180498

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