Properties

Label 2-276-23.4-c1-0-1
Degree $2$
Conductor $276$
Sign $0.403 + 0.914i$
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 + (−0.332 − 0.213i)5-s + (−0.440 − 3.06i)7-s + (0.841 − 0.540i)9-s + (1.09 − 2.39i)11-s + (0.315 − 2.19i)13-s + (0.379 + 0.111i)15-s + (2.42 + 2.80i)17-s + (4.61 − 5.32i)19-s + (1.28 + 2.81i)21-s + (−4.06 − 2.54i)23-s + (−2.01 − 4.40i)25-s + (−0.654 + 0.755i)27-s + (4.37 + 5.04i)29-s + (−2.22 − 0.652i)31-s + ⋯
L(s)  = 1  + (−0.553 + 0.162i)3-s + (−0.148 − 0.0955i)5-s + (−0.166 − 1.15i)7-s + (0.280 − 0.180i)9-s + (0.330 − 0.722i)11-s + (0.0874 − 0.608i)13-s + (0.0979 + 0.0287i)15-s + (0.589 + 0.679i)17-s + (1.05 − 1.22i)19-s + (0.280 + 0.613i)21-s + (−0.847 − 0.530i)23-s + (−0.402 − 0.881i)25-s + (−0.126 + 0.145i)27-s + (0.811 + 0.936i)29-s + (−0.398 − 0.117i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.403 + 0.914i$
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: \((2,\ 276,\ (\ :1/2),\ 0.403 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817115 - 0.532479i\)
\(L(\frac12)\) \(\approx\) \(0.817115 - 0.532479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.06 + 2.54i)T \)
good5 \( 1 + (0.332 + 0.213i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (0.440 + 3.06i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.09 + 2.39i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.315 + 2.19i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.42 - 2.80i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-4.61 + 5.32i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-4.37 - 5.04i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (2.22 + 0.652i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (4.80 - 3.08i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (2.36 + 1.51i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-3.75 + 1.10i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 4.37T + 47T^{2} \)
53 \( 1 + (-1.80 - 12.5i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.40 - 9.79i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-6.36 - 1.86i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-0.559 - 1.22i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (3.25 + 7.12i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-4.16 + 4.80i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (1.05 - 7.36i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (0.579 - 0.372i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-5.91 + 1.73i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (4.89 + 3.14i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.68013447039446863722349604155, −10.62651008369851825463512560497, −10.16420960107366785570698675333, −8.841843062731957853067065070876, −7.73350996341608733113691786197, −6.73675117907881770825079077506, −5.66743988639460200518038948391, −4.41441087247594653341113953494, −3.30891398304044271633918525841, −0.852784352194874807773019090405, 1.89518427844016702061147423965, 3.61030975435634333153222693123, 5.13519617649032309910533991383, 5.96697067675160439673402028283, 7.10678672610934286159760630904, 8.111827548661284738216399511451, 9.440712568631305115469713055743, 9.975072940949987284496719935648, 11.56075910910960056723129910940, 11.86959524195651691057290574593

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