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Properties

Label 2-231-77.76-c1-0-5
Degree $2$
Conductor $231$
Sign $0.206 - 0.978i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.206 - 0.978i$
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: \((2,\ 231,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08675 + 0.881261i\)
\(L(\frac12)\) \(\approx\) \(1.08675 + 0.881261i\)
\(L(\frac{3}{2})\) 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

−12.15766409685631612539405234270, −11.21947537380227144285850774242, −10.73077321971866053079498824275, −9.584515481437505732170398197675, −8.519983106510284963638405805127, −7.25435446850354874955239067888, −6.25336873803579435318959924814, −5.43250252191634437419566053896, −3.40994656879570467308101522616, −2.57917790247032179071831903350, 1.30557263337476017816154954390, 2.91373970782534047930272976854, 4.54999224125855369901844531602, 6.00847987845040173889693785260, 7.09016928196163703434281300069, 7.70228828244387513059211667681, 9.107895614164291879527675555117, 10.16739017305120097777166280946, 11.11487729886803981824821949503, 12.13164249462644548122003929137

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