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Properties

Label 2-276-69.44-c1-0-5
Degree $2$
Conductor $276$
Sign $0.995 + 0.0969i$
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  + (1.60 − 0.650i)3-s + (1.48 + 0.435i)5-s + (1.85 + 1.60i)7-s + (2.15 − 2.08i)9-s + (−4.36 + 2.80i)11-s + (−2.71 − 3.13i)13-s + (2.66 − 0.264i)15-s + (−1.86 + 4.07i)17-s + (7.30 − 3.33i)19-s + (4.02 + 1.37i)21-s + (−1.40 − 4.58i)23-s + (−2.20 − 1.41i)25-s + (2.10 − 4.75i)27-s + (−4.26 − 1.94i)29-s + (−1.13 + 7.87i)31-s + ⋯
L(s)  = 1  + (0.926 − 0.375i)3-s + (0.662 + 0.194i)5-s + (0.701 + 0.607i)7-s + (0.718 − 0.695i)9-s + (−1.31 + 0.845i)11-s + (−0.752 − 0.868i)13-s + (0.687 − 0.0683i)15-s + (−0.451 + 0.989i)17-s + (1.67 − 0.765i)19-s + (0.878 + 0.300i)21-s + (−0.293 − 0.956i)23-s + (−0.440 − 0.282i)25-s + (0.404 − 0.914i)27-s + (−0.792 − 0.362i)29-s + (−0.203 + 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.995 + 0.0969i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.995 + 0.0969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84343 - 0.0895535i\)
\(L(\frac12)\) \(\approx\) \(1.84343 - 0.0895535i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 0.650i)T \)
23 \( 1 + (1.40 + 4.58i)T \)
good5 \( 1 + (-1.48 - 0.435i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (-1.85 - 1.60i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (4.36 - 2.80i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.71 + 3.13i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (1.86 - 4.07i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-7.30 + 3.33i)T + (12.4 - 14.3i)T^{2} \)
29 \( 1 + (4.26 + 1.94i)T + (18.9 + 21.9i)T^{2} \)
31 \( 1 + (1.13 - 7.87i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.450 + 1.53i)T + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (-1.71 + 5.85i)T + (-34.4 - 22.1i)T^{2} \)
43 \( 1 + (-2.59 + 0.372i)T + (41.2 - 12.1i)T^{2} \)
47 \( 1 - 7.86iT - 47T^{2} \)
53 \( 1 + (2.42 - 2.80i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (4.23 - 3.66i)T + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (12.3 + 1.77i)T + (58.5 + 17.1i)T^{2} \)
67 \( 1 + (3.65 - 5.68i)T + (-27.8 - 60.9i)T^{2} \)
71 \( 1 + (2.38 - 3.71i)T + (-29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.48 - 3.25i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (0.00152 - 0.00131i)T + (11.2 - 78.1i)T^{2} \)
83 \( 1 + (-8.74 + 2.56i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-0.183 - 1.27i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-0.882 + 3.00i)T + (-81.6 - 52.4i)T^{2} \)
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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

−12.20956542787460790419975838489, −10.69869793409340963906721375823, −9.925528878865753843220478716958, −8.969692388431725539093407526570, −7.915062536412763037648796333355, −7.28314188060483470670006118824, −5.77918509292732800216233614810, −4.73508578544298869768168349330, −2.87220103372350144219486932149, −2.01042981157059322517307171875, 1.89739254636779493989219266996, 3.27028578427689338634576093466, 4.70242094931654819102686950790, 5.59350317627695315657373143449, 7.49291834901563053049665526583, 7.83116933567582902503657709024, 9.310765443431011551531771565781, 9.722439671496464739623294160069, 10.86193477079456459618169767289, 11.74917553266223250469594078392

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