Properties

Label 2-231-11.5-c1-0-8
Degree $2$
Conductor $231$
Sign $-0.113 + 0.993i$
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  + (−0.5 − 1.53i)2-s + (0.809 + 0.587i)3-s + (−0.5 + 0.363i)4-s + (0.190 − 0.587i)5-s + (0.5 − 1.53i)6-s + (0.809 − 0.587i)7-s + (−1.80 − 1.31i)8-s + (0.309 + 0.951i)9-s − 10-s + (2.80 − 1.76i)11-s − 0.618·12-s + (−0.0729 − 0.224i)13-s + (−1.30 − 0.951i)14-s + (0.5 − 0.363i)15-s + (−1.50 + 4.61i)16-s + (1.69 − 5.20i)17-s + ⋯
L(s)  = 1  + (−0.353 − 1.08i)2-s + (0.467 + 0.339i)3-s + (−0.250 + 0.181i)4-s + (0.0854 − 0.262i)5-s + (0.204 − 0.628i)6-s + (0.305 − 0.222i)7-s + (−0.639 − 0.464i)8-s + (0.103 + 0.317i)9-s − 0.316·10-s + (0.846 − 0.531i)11-s − 0.178·12-s + (−0.0202 − 0.0622i)13-s + (−0.349 − 0.254i)14-s + (0.129 − 0.0937i)15-s + (−0.375 + 1.15i)16-s + (0.410 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.113 + 0.993i$
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: \((2,\ 231,\ (\ :1/2),\ -0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854390 - 0.957320i\)
\(L(\frac12)\) \(\approx\) \(0.854390 - 0.957320i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 1.76i)T \)
good2 \( 1 + (0.5 + 1.53i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.0729 + 0.224i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.69 + 5.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2 - 1.45i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.70T + 23T^{2} \)
29 \( 1 + (3.42 - 2.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.927 - 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.85 - 3.52i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.54 - 1.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.52T + 43T^{2} \)
47 \( 1 + (-9.59 - 6.96i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.33 - 7.19i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.92 + 1.40i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.07 + 9.45i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + (4.16 - 12.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.78 - 5.65i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.57 - 4.84i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.145 - 0.449i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (5.78 + 17.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.76395015109755032290323997108, −10.99948900618568227910255342625, −9.967485312777577342558941487169, −9.311535848546917941859511877668, −8.383139908639715914282471479415, −7.03448817445642150582966527089, −5.56624524916865205402574191144, −4.02996426462646267312076079424, −2.92988363439295471638258207660, −1.34198471512577951866034080089, 2.16036572573081923361759851181, 3.90439555524014659031966272707, 5.65999297412013989086556398345, 6.57425292831900081127331056514, 7.48651650974909689029648566979, 8.333594927742506886912413292225, 9.167679652138246790770490067555, 10.28691015887542513831974395420, 11.73337021615127725125844523383, 12.34863743318889450248517413838

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