Properties

Label 2-276-12.11-c1-0-22
Degree $2$
Conductor $276$
Sign $0.301 - 0.953i$
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  + (1.40 + 0.146i)2-s + (−0.169 + 1.72i)3-s + (1.95 + 0.413i)4-s + 2.76i·5-s + (−0.492 + 2.39i)6-s − 3.82i·7-s + (2.69 + 0.868i)8-s + (−2.94 − 0.585i)9-s + (−0.406 + 3.89i)10-s + 2.92·11-s + (−1.04 + 3.30i)12-s − 4.70·13-s + (0.561 − 5.38i)14-s + (−4.76 − 0.469i)15-s + (3.65 + 1.61i)16-s + 4.06i·17-s + ⋯
L(s)  = 1  + (0.994 + 0.103i)2-s + (−0.0980 + 0.995i)3-s + (0.978 + 0.206i)4-s + 1.23i·5-s + (−0.200 + 0.979i)6-s − 1.44i·7-s + (0.951 + 0.307i)8-s + (−0.980 − 0.195i)9-s + (−0.128 + 1.23i)10-s + 0.883·11-s + (−0.301 + 0.953i)12-s − 1.30·13-s + (0.150 − 1.43i)14-s + (−1.23 − 0.121i)15-s + (0.914 + 0.404i)16-s + 0.985i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.301 - 0.953i$
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: \((2,\ 276,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75839 + 1.28816i\)
\(L(\frac12)\) \(\approx\) \(1.75839 + 1.28816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.146i)T \)
3 \( 1 + (0.169 - 1.72i)T \)
23 \( 1 + T \)
good5 \( 1 - 2.76iT - 5T^{2} \)
7 \( 1 + 3.82iT - 7T^{2} \)
11 \( 1 - 2.92T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 - 4.06iT - 17T^{2} \)
19 \( 1 + 3.75iT - 19T^{2} \)
29 \( 1 + 8.01iT - 29T^{2} \)
31 \( 1 + 1.15iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 1.54iT - 41T^{2} \)
43 \( 1 + 2.22iT - 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 6.30iT - 53T^{2} \)
59 \( 1 - 7.03T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 - 1.65iT - 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 5.68T + 73T^{2} \)
79 \( 1 - 3.06iT - 79T^{2} \)
83 \( 1 - 2.88T + 83T^{2} \)
89 \( 1 - 8.17iT - 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−11.82840054779655175472894366010, −11.12885043826742219456958091424, −10.40063950168248422690798464766, −9.699560681798489730963646866523, −7.84085211148508938402323642372, −6.91597945440058640371107635348, −6.09509575573344560092422951165, −4.53367474090575749372038293887, −3.89640699352924606759486140830, −2.71515072502769208720403985724, 1.59760686949204528089651730469, 2.85723375805134033820646905272, 4.78125889285171791312770380749, 5.48982680265511720967086565042, 6.48318437574042705858523256030, 7.65246610579502025488933267684, 8.732359399049489295124413219781, 9.663699167718258004285044302390, 11.50456011514454360369190680212, 11.99576690340289133102641610459

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