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Properties

Label 2-276-12.11-c1-0-20
Degree $2$
Conductor $276$
Sign $0.947 - 0.319i$
Analytic cond. $2.20387$
Root an. cond. $1.48454$
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  + (−1.37 + 0.346i)2-s + (1.70 + 0.293i)3-s + (1.75 − 0.950i)4-s − 2.20i·5-s + (−2.44 + 0.189i)6-s + 3.71i·7-s + (−2.08 + 1.91i)8-s + (2.82 + 1.00i)9-s + (0.762 + 3.01i)10-s + 0.958·11-s + (3.28 − 1.10i)12-s + 2.52·13-s + (−1.28 − 5.10i)14-s + (0.645 − 3.75i)15-s + (2.19 − 3.34i)16-s − 2.72i·17-s + ⋯
L(s)  = 1  + (−0.969 + 0.245i)2-s + (0.985 + 0.169i)3-s + (0.879 − 0.475i)4-s − 0.984i·5-s + (−0.997 + 0.0773i)6-s + 1.40i·7-s + (−0.736 + 0.676i)8-s + (0.942 + 0.333i)9-s + (0.241 + 0.954i)10-s + 0.288·11-s + (0.947 − 0.319i)12-s + 0.701·13-s + (−0.344 − 1.36i)14-s + (0.166 − 0.969i)15-s + (0.548 − 0.836i)16-s − 0.660i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19715 + 0.196276i\)
\(L(\frac12)\) \(\approx\) \(1.19715 + 0.196276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.346i)T \)
3 \( 1 + (-1.70 - 0.293i)T \)
23 \( 1 - T \)
good5 \( 1 + 2.20iT - 5T^{2} \)
7 \( 1 - 3.71iT - 7T^{2} \)
11 \( 1 - 0.958T + 11T^{2} \)
13 \( 1 - 2.52T + 13T^{2} \)
17 \( 1 + 2.72iT - 17T^{2} \)
19 \( 1 + 0.627iT - 19T^{2} \)
29 \( 1 - 6.59iT - 29T^{2} \)
31 \( 1 + 9.19iT - 31T^{2} \)
37 \( 1 + 8.67T + 37T^{2} \)
41 \( 1 + 5.91iT - 41T^{2} \)
43 \( 1 - 7.53iT - 43T^{2} \)
47 \( 1 + 4.76T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 0.0894T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 + 6.77iT - 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + 0.323iT - 89T^{2} \)
97 \( 1 - 7.00T + 97T^{2} \)
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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

−11.93900800461668880093854813676, −10.79589365961149943069088441778, −9.474519668533591841724082667070, −8.984427349506900854448000508710, −8.488794848911191542892395425494, −7.40580105409776526032498753011, −6.03331217391129126294475110937, −4.87664558831369622286403549474, −3.00338058868478630758225218620, −1.64175412284222656977549874254, 1.50422408252647144017850136312, 3.12477259356238436922426332479, 3.92918346421453537724711443989, 6.55653903294979136766701281425, 7.08201145492577922739082957233, 8.034670002081235943714433727166, 8.893441176492718522258514282392, 10.22426225230975235766299234317, 10.42723703158098663573524657693, 11.53215491008809007482021384586

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