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Properties

Label 2-276-12.11-c1-0-41
Degree $2$
Conductor $276$
Sign $-0.936 - 0.350i$
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  + (0.469 − 1.33i)2-s + (−1.64 + 0.542i)3-s + (−1.55 − 1.25i)4-s − 2.34i·5-s + (−0.0481 + 2.44i)6-s + 0.796i·7-s + (−2.40 + 1.49i)8-s + (2.41 − 1.78i)9-s + (−3.13 − 1.10i)10-s − 3.21·11-s + (3.24 + 1.21i)12-s − 6.69·13-s + (1.06 + 0.373i)14-s + (1.27 + 3.86i)15-s + (0.861 + 3.90i)16-s − 0.675i·17-s + ⋯
L(s)  = 1  + (0.331 − 0.943i)2-s + (−0.949 + 0.313i)3-s + (−0.779 − 0.626i)4-s − 1.05i·5-s + (−0.0196 + 0.999i)6-s + 0.300i·7-s + (−0.849 + 0.527i)8-s + (0.803 − 0.595i)9-s + (−0.990 − 0.348i)10-s − 0.969·11-s + (0.936 + 0.350i)12-s − 1.85·13-s + (0.283 + 0.0999i)14-s + (0.329 + 0.997i)15-s + (0.215 + 0.976i)16-s − 0.163i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.936 - 0.350i$
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: \((2,\ 276,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0872562 + 0.482136i\)
\(L(\frac12)\) \(\approx\) \(0.0872562 + 0.482136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.469 + 1.33i)T \)
3 \( 1 + (1.64 - 0.542i)T \)
23 \( 1 + T \)
good5 \( 1 + 2.34iT - 5T^{2} \)
7 \( 1 - 0.796iT - 7T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 + 6.69T + 13T^{2} \)
17 \( 1 + 0.675iT - 17T^{2} \)
19 \( 1 + 0.720iT - 19T^{2} \)
29 \( 1 + 1.92iT - 29T^{2} \)
31 \( 1 + 7.61iT - 31T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 1.22T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 - 0.176T + 59T^{2} \)
61 \( 1 - 6.71T + 61T^{2} \)
67 \( 1 + 9.36iT - 67T^{2} \)
71 \( 1 + 9.44T + 71T^{2} \)
73 \( 1 - 0.422T + 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + 8.44T + 83T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + 7.27T + 97T^{2} \)
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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

−11.55822313012009742757961069693, −10.44335903797369189653095438438, −9.754124498924084412859052524385, −8.887012468039968353832872599913, −7.44606480385265475088409776814, −5.70416485172518850418449636132, −5.08069405879847794764256066493, −4.25679526061271444790275217556, −2.36870145581529130979667977509, −0.36007931294911492424113104173, 2.84525133044849954887350805300, 4.59246187011366522851426474726, 5.44305596929445776698743928270, 6.64538099295993878051571917930, 7.23344350823876230457475554280, 8.037362427852403086925450054545, 9.804945548490895641301620330305, 10.45268830977248082973906648114, 11.60245119023482988394294608996, 12.55772962339561235460788515307

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