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Properties

Label 2-276-276.275-c0-0-5
Degree $2$
Conductor $276$
Sign $-0.5 + 0.866i$
Analytic cond. $0.137741$
Root an. cond. $0.371136$
Motivic weight $0$
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 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + 23-s + (0.499 + 0.866i)24-s − 25-s + (0.5 − 0.866i)26-s + 0.999·27-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + 23-s + (0.499 + 0.866i)24-s − 25-s + (0.5 − 0.866i)26-s + 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(0.137741\)
Root analytic conductor: \(0.371136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7851153577\)
\(L(\frac12)\) \(\approx\) \(0.7851153577\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.76729686910859802765950955344, −11.16089822397120725876836697606, −10.26610519469418080074141594019, −9.035389408502922851715141576206, −7.945925631455983445365590095950, −6.53815911077038375973259754830, −5.74849852753773360391225925447, −4.53947872496017580081189961752, −3.01331157659062557389038820446, −1.47366450427510875911822042463, 3.32914726616853574232908607697, 4.31622591088527698969058230147, 5.45020930768995428672893503353, 6.22178778357482623339037209118, 7.39493287008180899325295428374, 8.680969507768690886937833454751, 9.357633962448332351498446881115, 10.65070831833478680064154049601, 11.50553706777584285060768554109, 12.48595537714980404353599283399

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