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Properties

Label 2-276-276.275-c0-0-1
Degree $2$
Conductor $276$
Sign $1$
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 + (0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·12-s + 13-s + (−0.5 − 0.866i)16-s + 0.999·18-s − 23-s + (0.5 + 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 + (0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·12-s + 13-s + (−0.5 − 0.866i)16-s + 0.999·18-s − 23-s + (0.5 + 0.866i)24-s − 25-s + (−0.5 − 0.866i)26-s − 0.999·27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $1$
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: \((2,\ 276,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6692191272\)
\(L(\frac12)\) \(\approx\) \(0.6692191272\)
\(L(1)\) 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

−11.63075216399526877768610488667, −11.23605330536679643242520830880, −9.938337436989471838831515276625, −9.655080417764662182485215472039, −8.358998276420851463323738329659, −7.88036979286137812645767799563, −5.99130813376410036201674637656, −4.41958513492956477317319727359, −3.60892462546021914698239996602, −2.21543474792341932808993074555, 1.64425333060033344749402799769, 3.65993122708462194599119316364, 5.40028989208129575099032959000, 6.43976013446279871700156380824, 7.23666090356560335081352254691, 8.308550141957662946975868327424, 8.833721293587256502114122514770, 9.982084605689207700190134519129, 11.07051760291761965682653486424, 12.28369280332853424723524056946

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