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Properties

Label 2-276-23.6-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: $\chi_{276} (121, \cdot )$
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

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

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