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Properties

Label 2-231-7.4-c1-0-6
Degree $2$
Conductor $231$
Sign $0.386 - 0.922i$
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 + 1.73i)2-s + (0.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + 1.99·6-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (1 + 1.73i)12-s − 5·13-s + (1.00 + 5.19i)14-s + (1.99 + 3.46i)16-s + (−3 + 5.19i)17-s + (0.999 − 1.73i)18-s + (−3.5 − 6.06i)19-s + (2 − 1.73i)21-s + 1.99·22-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (0.288 − 0.499i)3-s + (−0.499 + 0.866i)4-s + 0.816·6-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s + (0.288 + 0.499i)12-s − 1.38·13-s + (0.267 + 1.38i)14-s + (0.499 + 0.866i)16-s + (−0.727 + 1.26i)17-s + (0.235 − 0.408i)18-s + (−0.802 − 1.39i)19-s + (0.436 − 0.377i)21-s + 0.426·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63887 + 1.09015i\)
\(L(\frac12)\) \(\approx\) \(1.63887 + 1.09015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−12.76330192244983962234379002223, −11.64364226971689673781274977003, −10.57224368012032384238839533815, −8.966061030100122121943057064239, −8.142826525168905711157897447341, −7.23332851648736729301646181171, −6.35770667042513005377017322834, −5.19233923583365002529897989329, −4.26179664777318254873576045482, −2.21592468786932712240342476490, 1.89986761739282675830071919553, 3.16081551750051396315896111487, 4.58673912664646542633913166884, 4.97024898212468572981284269089, 7.04669903137814679268932128991, 8.183737458587139982514204139339, 9.502913131229102695737461291191, 10.35844163870448281900821259203, 11.12544473443726703312103083774, 11.99056174492223774297945854722

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