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Properties

Label 2-276-23.4-c1-0-3
Degree $2$
Conductor $276$
Sign $-0.0868 + 0.996i$
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  + (0.959 − 0.281i)3-s + (−3.55 − 2.28i)5-s + (0.00534 + 0.0371i)7-s + (0.841 − 0.540i)9-s + (2.38 − 5.22i)11-s + (0.217 − 1.51i)13-s + (−4.05 − 1.19i)15-s + (−3.33 − 3.85i)17-s + (−0.384 + 0.444i)19-s + (0.0155 + 0.0341i)21-s + (2.33 + 4.18i)23-s + (5.33 + 11.6i)25-s + (0.654 − 0.755i)27-s + (2.90 + 3.35i)29-s + (−6.02 − 1.76i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.162i)3-s + (−1.58 − 1.02i)5-s + (0.00201 + 0.0140i)7-s + (0.280 − 0.180i)9-s + (0.718 − 1.57i)11-s + (0.0602 − 0.419i)13-s + (−1.04 − 0.307i)15-s + (−0.809 − 0.934i)17-s + (−0.0883 + 0.101i)19-s + (0.00340 + 0.00744i)21-s + (0.487 + 0.873i)23-s + (1.06 + 2.33i)25-s + (0.126 − 0.145i)27-s + (0.539 + 0.623i)29-s + (−1.08 − 0.317i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0868 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0868 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.0868 + 0.996i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.0868 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735684 - 0.802588i\)
\(L(\frac12)\) \(\approx\) \(0.735684 - 0.802588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-2.33 - 4.18i)T \)
good5 \( 1 + (3.55 + 2.28i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.00534 - 0.0371i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-2.38 + 5.22i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.217 + 1.51i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (3.33 + 3.85i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.384 - 0.444i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-2.90 - 3.35i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (6.02 + 1.76i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (0.00692 - 0.00445i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-7.48 - 4.81i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-5.23 + 1.53i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 8.73T + 47T^{2} \)
53 \( 1 + (-0.0960 - 0.668i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.44 + 10.0i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-14.1 - 4.15i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-1.71 - 3.76i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-0.566 - 1.23i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-1.23 + 1.42i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.122 + 0.854i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-4.29 + 2.75i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (11.5 - 3.40i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-3.28 - 2.11i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.50638661552792634908571808246, −11.14669768565158945883935856151, −9.299818026335131204661728664258, −8.683832286638455059188310491421, −7.953934120208811267199175596371, −6.96484031778317899618625652993, −5.38260490889415792708183251792, −4.11669641382647100106516774681, −3.23663609564718961337398167330, −0.825638726791926320768978989563, 2.39277499868043774373241934543, 3.89399196620299901322345560051, 4.40646731393497845668311586855, 6.64593803228247179894314591340, 7.20995315590260843925505740363, 8.197935765613921984241878154270, 9.193778854554497168461153725529, 10.40495820000431398504216368588, 11.14008315209172069018608930044, 12.10704985324590851848663804485

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