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Properties

Label 2-276-4.3-c2-0-37
Degree $2$
Conductor $276$
Sign $-0.996 - 0.0784i$
Analytic cond. $7.52045$
Root an. cond. $2.74234$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 0.0784i)2-s − 1.73i·3-s + (3.98 + 0.313i)4-s − 0.934·5-s + (−0.135 + 3.46i)6-s − 12.0i·7-s + (−7.94 − 0.939i)8-s − 2.99·9-s + (1.86 + 0.0733i)10-s + 10.1i·11-s + (0.543 − 6.90i)12-s − 17.9·13-s + (−0.948 + 24.1i)14-s + 1.61i·15-s + (15.8 + 2.50i)16-s + 21.4·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0392i)2-s − 0.577i·3-s + (0.996 + 0.0784i)4-s − 0.186·5-s + (−0.0226 + 0.576i)6-s − 1.72i·7-s + (−0.993 − 0.117i)8-s − 0.333·9-s + (0.186 + 0.00733i)10-s + 0.921i·11-s + (0.0452 − 0.575i)12-s − 1.37·13-s + (−0.0677 + 1.72i)14-s + 0.107i·15-s + (0.987 + 0.156i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.996 - 0.0784i$
Analytic conductor: \(7.52045\)
Root analytic conductor: \(2.74234\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1),\ -0.996 - 0.0784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0155462 + 0.395821i\)
\(L(\frac12)\) \(\approx\) \(0.0155462 + 0.395821i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 0.0784i)T \)
3 \( 1 + 1.73iT \)
23 \( 1 - 4.79iT \)
good5 \( 1 + 0.934T + 25T^{2} \)
7 \( 1 + 12.0iT - 49T^{2} \)
11 \( 1 - 10.1iT - 121T^{2} \)
13 \( 1 + 17.9T + 169T^{2} \)
17 \( 1 - 21.4T + 289T^{2} \)
19 \( 1 + 25.2iT - 361T^{2} \)
29 \( 1 + 41.8T + 841T^{2} \)
31 \( 1 - 52.6iT - 961T^{2} \)
37 \( 1 + 23.7T + 1.36e3T^{2} \)
41 \( 1 + 26.7T + 1.68e3T^{2} \)
43 \( 1 + 19.3iT - 1.84e3T^{2} \)
47 \( 1 + 28.2iT - 2.20e3T^{2} \)
53 \( 1 + 90.8T + 2.80e3T^{2} \)
59 \( 1 + 27.6iT - 3.48e3T^{2} \)
61 \( 1 - 54.4T + 3.72e3T^{2} \)
67 \( 1 + 96.3iT - 4.48e3T^{2} \)
71 \( 1 - 68.0iT - 5.04e3T^{2} \)
73 \( 1 - 122.T + 5.32e3T^{2} \)
79 \( 1 + 65.9iT - 6.24e3T^{2} \)
83 \( 1 + 28.4iT - 6.88e3T^{2} \)
89 \( 1 + 86.0T + 7.92e3T^{2} \)
97 \( 1 - 117.T + 9.40e3T^{2} \)
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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

−11.01928407865387766067342798322, −10.10855098799096463667830108167, −9.477477341943060435711544415248, −7.976611218010327144845602277348, −7.26176281995300732745987735018, −6.89713065796675734301017354782, −5.05951960121398681089239524925, −3.42309598281791142708200421263, −1.75483510524616138656979748399, −0.25941147561033620197128933386, 2.17004145014280690186916424089, 3.39113200521050518985786149400, 5.46777139378216296370092570245, 5.98229910133654081722515978388, 7.71439662928796034459395220496, 8.342686309634568637759592621843, 9.498723828529590076689255442570, 9.847785216523056228361791201136, 11.21205635015646846586099452158, 11.90534352988669530293713233666

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