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Properties

Label 2-231-231.32-c1-0-2
Degree $2$
Conductor $231$
Sign $0.205 - 0.978i$
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 + (−3.05 + 1.29i)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 + (−0.920 + 0.391i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265519 + 0.215614i\)
\(L(\frac12)\) \(\approx\) \(0.265519 + 0.215614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.05 - 1.29i)T \)
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

−11.96538585841398294514593850311, −11.36017755224425046188863083716, −10.14609498693403489112811064598, −9.784388480849934018657309033989, −9.120255229535977536392638599820, −7.986226743994297702113768917611, −6.20704092468422636519638440415, −4.58224512818005230776503758812, −3.45695631924108238009341563125, −2.22537245862705984571047639009, 0.34751244722985290151697205803, 2.84810477131340703733665604871, 5.45626117814580329050835632105, 6.15979325070090488867886682409, 6.91954148771547692899834114965, 8.134519750866685399805904647127, 8.414607520750264067796541211616, 9.954199893196132074442260634841, 10.52518731374796532401907217641, 12.32875121302181912047335708116

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