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Properties

Label 2-276-92.91-c1-0-2
Degree $2$
Conductor $276$
Sign $-0.836 + 0.548i$
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  + (−0.588 + 1.28i)2-s + i·3-s + (−1.30 − 1.51i)4-s + 2.28i·5-s + (−1.28 − 0.588i)6-s − 3.41·7-s + (2.71 − 0.792i)8-s − 9-s + (−2.94 − 1.34i)10-s − 1.67·11-s + (1.51 − 1.30i)12-s + 0.215·13-s + (2.00 − 4.39i)14-s − 2.28·15-s + (−0.577 + 3.95i)16-s + 0.222i·17-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + 0.577i·3-s + (−0.654 − 0.756i)4-s + 1.02i·5-s + (−0.525 − 0.240i)6-s − 1.29·7-s + (0.959 − 0.280i)8-s − 0.333·9-s + (−0.930 − 0.425i)10-s − 0.503·11-s + (0.436 − 0.377i)12-s + 0.0598·13-s + (0.536 − 1.17i)14-s − 0.590·15-s + (−0.144 + 0.989i)16-s + 0.0540i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.836 + 0.548i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135938 - 0.455124i\)
\(L(\frac12)\) \(\approx\) \(0.135938 - 0.455124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.588 - 1.28i)T \)
3 \( 1 - iT \)
23 \( 1 + (-4.75 - 0.633i)T \)
good5 \( 1 - 2.28iT - 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 - 0.215T + 13T^{2} \)
17 \( 1 - 0.222iT - 17T^{2} \)
19 \( 1 + 7.32T + 19T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 3.33iT - 31T^{2} \)
37 \( 1 - 1.86iT - 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 2.67T + 43T^{2} \)
47 \( 1 - 6.03iT - 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 - 3.93iT - 59T^{2} \)
61 \( 1 - 9.89iT - 61T^{2} \)
67 \( 1 - 1.94T + 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 4.15T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−12.71774942722678487300949059529, −10.91424065479653114264102508794, −10.49678708681716845805345820237, −9.534290308669242976418787513832, −8.728995305439016808231842711620, −7.39760364219002818590109692306, −6.56215721257133177337842538743, −5.74801859129006544102472972617, −4.25662930085245450604768124293, −2.90147123734985531490424219839, 0.39808952510235213785980389784, 2.21236420495690543557340339478, 3.61857017106785189747504049418, 4.97737319873538930727041194237, 6.40399874419403280583031473183, 7.66580474069346942122741288685, 8.725876951702307665040000091845, 9.320437452002714898556483794304, 10.39498133580975473100990636922, 11.35670814302224299480534338570

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