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Properties

Label 2-231-11.5-c1-0-3
Degree $2$
Conductor $231$
Sign $-0.836 - 0.548i$
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  + (0.690 + 2.12i)2-s + (0.809 + 0.587i)3-s + (−2.42 + 1.76i)4-s + (−0.190 + 0.587i)5-s + (−0.690 + 2.12i)6-s + (−0.809 + 0.587i)7-s + (−1.80 − 1.31i)8-s + (0.309 + 0.951i)9-s − 1.38·10-s + (−0.309 − 3.30i)11-s − 3·12-s + (1 + 3.07i)13-s + (−1.80 − 1.31i)14-s + (−0.5 + 0.363i)15-s + (−0.309 + 0.951i)16-s + (1.5 − 4.61i)17-s + ⋯
L(s)  = 1  + (0.488 + 1.50i)2-s + (0.467 + 0.339i)3-s + (−1.21 + 0.881i)4-s + (−0.0854 + 0.262i)5-s + (−0.282 + 0.868i)6-s + (−0.305 + 0.222i)7-s + (−0.639 − 0.464i)8-s + (0.103 + 0.317i)9-s − 0.437·10-s + (−0.0931 − 0.995i)11-s − 0.866·12-s + (0.277 + 0.853i)13-s + (−0.483 − 0.351i)14-s + (−0.129 + 0.0937i)15-s + (−0.0772 + 0.237i)16-s + (0.363 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.836 - 0.548i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477926 + 1.59931i\)
\(L(\frac12)\) \(\approx\) \(0.477926 + 1.59931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 3.30i)T \)
good2 \( 1 + (-0.690 - 2.12i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.190 - 0.587i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1 - 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.5 + 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.30 + 1.67i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.38T + 23T^{2} \)
29 \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.954 + 2.93i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.73 - 2.71i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.97 + 4.33i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + (3.61 + 2.62i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.09 - 6.43i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.61 - 1.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (1.52 - 4.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.0 - 8.05i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.09 + 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.09 + 15.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + (1.85 + 5.70i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.13966610672071907620996229068, −11.69629030631547376921013513245, −10.62947393073690279624463434588, −9.158153676317000081767146347384, −8.595524802711426334149646827398, −7.38802888192933160720681712939, −6.58081179938490770254020466968, −5.48250980032149488080858725882, −4.38119761209234627735904029778, −3.02717827356592508391798936121, 1.38398853279145817269965663805, 2.82617140227775771838161795188, 3.90352337734420169249135108571, 5.07557373058760816558962433311, 6.71850506977507369840778928231, 8.044853697327831252247979152892, 9.135389100674741248359182485741, 10.28738970474745716840315951701, 10.70995920984484185610805051302, 12.18029923766736325481225889029

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