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Properties

Label 2-231-231.32-c1-0-13
Degree $2$
Conductor $231$
Sign $-0.465 + 0.885i$
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  + (−1.36 − 2.36i)2-s + (1.68 − 0.379i)3-s + (−2.72 + 4.72i)4-s + (1.96 − 1.13i)5-s + (−3.20 − 3.47i)6-s + (1.90 + 1.83i)7-s + 9.44·8-s + (2.71 − 1.28i)9-s + (−5.37 − 3.10i)10-s + (−2.00 − 2.64i)11-s + (−2.81 + 9.02i)12-s − 0.322i·13-s + (1.74 − 7.01i)14-s + (2.89 − 2.66i)15-s + (−7.43 − 12.8i)16-s + (−1.73 + 2.99i)17-s + ⋯
L(s)  = 1  + (−0.965 − 1.67i)2-s + (0.975 − 0.219i)3-s + (−1.36 + 2.36i)4-s + (0.880 − 0.508i)5-s + (−1.30 − 1.42i)6-s + (0.719 + 0.694i)7-s + 3.33·8-s + (0.903 − 0.427i)9-s + (−1.69 − 0.981i)10-s + (−0.604 − 0.796i)11-s + (−0.813 + 2.60i)12-s − 0.0894i·13-s + (0.465 − 1.87i)14-s + (0.747 − 0.688i)15-s + (−1.85 − 3.21i)16-s + (−0.419 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.465 + 0.885i$
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: \((2,\ 231,\ (\ :1/2),\ -0.465 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.595801 - 0.986395i\)
\(L(\frac12)\) \(\approx\) \(0.595801 - 0.986395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 + 0.379i)T \)
7 \( 1 + (-1.90 - 1.83i)T \)
11 \( 1 + (2.00 + 2.64i)T \)
good2 \( 1 + (1.36 + 2.36i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.96 + 1.13i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.322iT - 13T^{2} \)
17 \( 1 + (1.73 - 2.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.16 + 1.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.67 - 1.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.690 + 1.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.42T + 41T^{2} \)
43 \( 1 + 5.82iT - 43T^{2} \)
47 \( 1 + (-2.37 + 1.37i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.824 + 0.475i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.17 - 5.29i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0609 - 0.0351i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.262 + 0.455i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.71iT - 71T^{2} \)
73 \( 1 + (-3.18 - 1.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.08 - 2.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.50T + 83T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−11.78751629052837043118358250475, −10.80523957823333923219231593726, −9.849632847219691653281683836515, −8.979536605840301986860075130880, −8.535312058580279718175404330436, −7.56958780194544629893639065421, −5.31466078677094755810103326510, −3.71046328150833494092538927868, −2.43814511138067276482884643215, −1.53890691037714953285288240546, 1.90953885539624247091957689679, 4.43295068462920888110867329015, 5.52845394058662104952399574088, 6.91980137150711523506966030104, 7.55431523952570121882656022864, 8.354330504468515060487893144031, 9.625648035001303295529639342554, 9.891096301187452724924914210092, 10.90829537250684136279879130532, 13.23297463488098646479582903321

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