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Properties

Label 2-231-77.76-c1-0-6
Degree $2$
Conductor $231$
Sign $0.825 - 0.563i$
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  − 0.183i·2-s + i·3-s + 1.96·4-s + 1.83i·5-s + 0.183·6-s + (1.47 − 2.19i)7-s − 0.726i·8-s − 9-s + 0.335·10-s + (−1.23 + 3.07i)11-s + 1.96i·12-s − 0.879·13-s + (−0.402 − 0.269i)14-s − 1.83·15-s + 3.79·16-s + 3.57·17-s + ⋯
L(s)  = 1  − 0.129i·2-s + 0.577i·3-s + 0.983·4-s + 0.819i·5-s + 0.0747·6-s + (0.556 − 0.830i)7-s − 0.256i·8-s − 0.333·9-s + 0.106·10-s + (−0.372 + 0.927i)11-s + 0.567i·12-s − 0.243·13-s + (−0.107 − 0.0720i)14-s − 0.473·15-s + 0.949·16-s + 0.867·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.825 - 0.563i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.825 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44742 + 0.446879i\)
\(L(\frac12)\) \(\approx\) \(1.44742 + 0.446879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-1.47 + 2.19i)T \)
11 \( 1 + (1.23 - 3.07i)T \)
good2 \( 1 + 0.183iT - 2T^{2} \)
5 \( 1 - 1.83iT - 5T^{2} \)
13 \( 1 + 0.879T + 13T^{2} \)
17 \( 1 - 3.57T + 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 - 0.539T + 23T^{2} \)
29 \( 1 + 3.82iT - 29T^{2} \)
31 \( 1 + 5.59iT - 31T^{2} \)
37 \( 1 + 2.63T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 6.21iT - 43T^{2} \)
47 \( 1 + 5.29iT - 47T^{2} \)
53 \( 1 + 3.93T + 53T^{2} \)
59 \( 1 + 10.2iT - 59T^{2} \)
61 \( 1 - 0.732T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 11.3iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 2.80iT - 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−12.02711962100053645306382883989, −11.13677973387597883015430749805, −10.41355538799314021938413422236, −9.889263200266155756857285979054, −8.108453605991632475966861145446, −7.26139507117850821297774596875, −6.34529733002965947377230851980, −4.83477203260700009799653727880, −3.51580468600874139816807838258, −2.14134406153041347560310488633, 1.59573851592133131692039330499, 2.99647186779452091291857601714, 5.10007292836399919336309645006, 5.92070647183938196034727205213, 7.07959117200185086240956206914, 8.294020218693837040457499264623, 8.715465701337211386775566742247, 10.41022768070370090414127796273, 11.30441948642710840032710091589, 12.25585484906510694098446699281

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