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Properties

Label 2-276-23.3-c1-0-2
Degree $2$
Conductor $276$
Sign $0.995 - 0.0927i$
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  + (−0.415 + 0.909i)3-s + (0.542 − 0.625i)5-s + (3.79 − 2.43i)7-s + (−0.654 − 0.755i)9-s + (−0.421 + 2.92i)11-s + (−2.52 − 1.62i)13-s + (0.344 + 0.753i)15-s + (5.65 + 1.66i)17-s + (5.36 − 1.57i)19-s + (0.641 + 4.45i)21-s + (4.33 + 2.04i)23-s + (0.613 + 4.27i)25-s + (0.959 − 0.281i)27-s + (−9.91 − 2.91i)29-s + (−0.0738 − 0.161i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (0.242 − 0.279i)5-s + (1.43 − 0.920i)7-s + (−0.218 − 0.251i)9-s + (−0.126 + 0.883i)11-s + (−0.700 − 0.449i)13-s + (0.0888 + 0.194i)15-s + (1.37 + 0.402i)17-s + (1.22 − 0.361i)19-s + (0.139 + 0.973i)21-s + (0.904 + 0.427i)23-s + (0.122 + 0.854i)25-s + (0.184 − 0.0542i)27-s + (−1.84 − 0.540i)29-s + (−0.0132 − 0.0290i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.995 - 0.0927i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.995 - 0.0927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38554 + 0.0643780i\)
\(L(\frac12)\) \(\approx\) \(1.38554 + 0.0643780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-4.33 - 2.04i)T \)
good5 \( 1 + (-0.542 + 0.625i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-3.79 + 2.43i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.421 - 2.92i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (2.52 + 1.62i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-5.65 - 1.66i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-5.36 + 1.57i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (9.91 + 2.91i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.0738 + 0.161i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (6.71 + 7.75i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (5.72 - 6.60i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-0.230 + 0.503i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 4.82T + 47T^{2} \)
53 \( 1 + (4.70 - 3.02i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-0.611 - 0.392i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (1.09 + 2.40i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.100 - 0.698i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.31 + 9.15i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-0.159 + 0.0469i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (1.36 + 0.874i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (1.81 + 2.09i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (0.471 - 1.03i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (9.45 - 10.9i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.72104866954749510006926865924, −10.95259545092717749086817111657, −10.02900693541897777016055326539, −9.272882170906256422499034391784, −7.76131600577312940287249850042, −7.34624848621982205182935802541, −5.36937882517156564304697956810, −4.94437685428719365089285912032, −3.56670670726736351625068817289, −1.49645990996093800570543173651, 1.61440325270761794915398222323, 3.05359522003052458711983151055, 5.09756023169985613984965660525, 5.60550763327580927621065514386, 7.05103730619964623586619781623, 7.976902528959285340541727108098, 8.837710178116659065144424183972, 10.04178817754528894717374595619, 11.24485082036075521638354018330, 11.76892457694021677143331673337

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