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Properties

Label 2-231-11.9-c1-0-8
Degree $2$
Conductor $231$
Sign $0.569 + 0.821i$
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  + (0.0501 − 0.154i)2-s + (0.809 − 0.587i)3-s + (1.59 + 1.16i)4-s + (−1.35 − 4.17i)5-s + (−0.0501 − 0.154i)6-s + (0.809 + 0.587i)7-s + (0.521 − 0.378i)8-s + (0.309 − 0.951i)9-s − 0.711·10-s + (−3.30 − 0.224i)11-s + 1.97·12-s + (0.517 − 1.59i)13-s + (0.131 − 0.0953i)14-s + (−3.54 − 2.57i)15-s + (1.18 + 3.65i)16-s + (1.91 + 5.88i)17-s + ⋯
L(s)  = 1  + (0.0354 − 0.109i)2-s + (0.467 − 0.339i)3-s + (0.798 + 0.580i)4-s + (−0.606 − 1.86i)5-s + (−0.0204 − 0.0629i)6-s + (0.305 + 0.222i)7-s + (0.184 − 0.133i)8-s + (0.103 − 0.317i)9-s − 0.224·10-s + (−0.997 − 0.0676i)11-s + 0.569·12-s + (0.143 − 0.442i)13-s + (0.0350 − 0.0254i)14-s + (−0.916 − 0.665i)15-s + (0.296 + 0.913i)16-s + (0.463 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37058 - 0.717434i\)
\(L(\frac12)\) \(\approx\) \(1.37058 - 0.717434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.30 + 0.224i)T \)
good2 \( 1 + (-0.0501 + 0.154i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.35 + 4.17i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.517 + 1.59i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.91 - 5.88i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.81 + 2.04i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.568T + 23T^{2} \)
29 \( 1 + (-7.17 - 5.21i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.33 + 4.12i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.784 + 0.569i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.67 - 3.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.04T + 43T^{2} \)
47 \( 1 + (3.78 - 2.74i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.735 - 2.26i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.30 - 1.67i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.87 - 8.84i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.14T + 67T^{2} \)
71 \( 1 + (0.245 + 0.755i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.93 + 1.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.207 - 0.637i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.86 + 8.83i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + (-4.65 + 14.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.26043156743988917348502859730, −11.41402070783388945953073514529, −10.12975106543728721056271058193, −8.584333401996129812441155824949, −8.272450948801096802165111619346, −7.42261907503427038182998664488, −5.75984640817633293728057703460, −4.56417196363252836840438788224, −3.18183259390999204339182749379, −1.46215210082188630024293097376, 2.46769919797971047542598755295, 3.33930832814439897763672804299, 5.06378885468013189214501644365, 6.53062467545028499031610492909, 7.29923614324098041471613054921, 8.039090118907384772166710337446, 9.969086018371152643496048136450, 10.31657550178370206219079284315, 11.33258375027962185984505065681, 11.85557502466661494238470101623

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