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Properties

Label 2-276-92.91-c1-0-14
Degree $2$
Conductor $276$
Sign $0.135 + 0.990i$
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  + (−1.36 + 0.377i)2-s − i·3-s + (1.71 − 1.02i)4-s − 2.63i·5-s + (0.377 + 1.36i)6-s + 3.76·7-s + (−1.94 + 2.04i)8-s − 9-s + (0.995 + 3.59i)10-s − 0.443·11-s + (−1.02 − 1.71i)12-s − 3.89·13-s + (−5.12 + 1.42i)14-s − 2.63·15-s + (1.88 − 3.52i)16-s − 5.41i·17-s + ⋯
L(s)  = 1  + (−0.963 + 0.266i)2-s − 0.577i·3-s + (0.857 − 0.514i)4-s − 1.17i·5-s + (0.154 + 0.556i)6-s + 1.42·7-s + (−0.689 + 0.724i)8-s − 0.333·9-s + (0.314 + 1.13i)10-s − 0.133·11-s + (−0.296 − 0.495i)12-s − 1.08·13-s + (−1.37 + 0.379i)14-s − 0.680·15-s + (0.470 − 0.882i)16-s − 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.135 + 0.990i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663074 - 0.578407i\)
\(L(\frac12)\) \(\approx\) \(0.663074 - 0.578407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.377i)T \)
3 \( 1 + iT \)
23 \( 1 + (-4.40 - 1.88i)T \)
good5 \( 1 + 2.63iT - 5T^{2} \)
7 \( 1 - 3.76T + 7T^{2} \)
11 \( 1 + 0.443T + 11T^{2} \)
13 \( 1 + 3.89T + 13T^{2} \)
17 \( 1 + 5.41iT - 17T^{2} \)
19 \( 1 - 0.874T + 19T^{2} \)
29 \( 1 + 5.34T + 29T^{2} \)
31 \( 1 + 0.598iT - 31T^{2} \)
37 \( 1 + 8.26iT - 37T^{2} \)
41 \( 1 - 3.00T + 41T^{2} \)
43 \( 1 - 0.586T + 43T^{2} \)
47 \( 1 + 3.89iT - 47T^{2} \)
53 \( 1 - 3.21iT - 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 - 1.61iT - 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 - 1.29iT - 97T^{2} \)
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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

−11.67256217615111970602297982491, −10.78216865581710891487598021107, −9.362688377048042068359869762298, −8.836574230556301534284199591908, −7.68141121720543254858448968194, −7.30662731253295769972703425905, −5.52020463783600047439373715767, −4.86747313744339153109180037536, −2.29723121822724054781294227628, −0.963824112435216258471681472562, 2.03386786685138704439606286552, 3.32877251232603296063742104453, 4.87009449157446962627850763854, 6.39402508486793561008312262687, 7.51536120168000861045055813315, 8.226133389145051690237861698323, 9.385823218747660852672086242349, 10.40150063439737727861952221896, 10.92973271525539184131685981393, 11.58653833287542424872732970482

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