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Properties

Label 2-276-12.11-c1-0-30
Degree $2$
Conductor $276$
Sign $-0.544 + 0.839i$
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 + (−1.49 − 1.93i)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 + (−1.88 + 2.90i)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 + (−0.610 − 0.791i)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 + (−0.544 + 0.839i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.544 + 0.839i$
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: \((2,\ 276,\ (\ :1/2),\ -0.544 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641144 - 1.17981i\)
\(L(\frac12)\) \(\approx\) \(0.641144 - 1.17981i\)
\(L(\frac{3}{2})\) 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

−11.94170707558196632325111895194, −10.40672453203779014515262173699, −9.437535338196409635965456125257, −8.994822154340252046379357377397, −7.948546110406144916005663522200, −7.04732935461096361397124758983, −4.93849331906985052065007267674, −4.02396140101546566629162155418, −2.60191559885088535218375447008, −1.18143137840875062606284336450, 2.38621035153833814798776988763, 3.89794135974920332387937212544, 5.13925907665224427017426530196, 6.66484692072822515113569716240, 7.28769345905426683964884659923, 8.273534289672454762524155315713, 9.415792330244999406190874507820, 9.858404096145459446636222691495, 10.94676839189899198602208710924, 12.33379869705577255663833632252

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