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Properties

Label 2-276-92.91-c1-0-16
Degree $2$
Conductor $276$
Sign $-0.810 + 0.586i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.810 + 0.586i$
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: \((2,\ 276,\ (\ :1/2),\ -0.810 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0956916 - 0.295442i\)
\(L(\frac12)\) \(\approx\) \(0.0956916 - 0.295442i\)
\(L(\frac{3}{2})\) 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

−11.44250674604090275538962165627, −10.20647084355142357704683691938, −9.424699575963106647928395034807, −9.125218683761280842599465063495, −7.82635292745328135511354627018, −6.72140087468205470010434892719, −5.41444480716955507286227400851, −3.97883725885605729875089562786, −2.57669854426502058091943453985, −0.29380815004306803011497336876, 2.21878658038813011490811935919, 3.39341414940807518998648043287, 5.86250543747583470520155050652, 6.60379105913028479785895804393, 7.25368953299780469110807602709, 8.297251517054551809632346425063, 9.631116199323547017271779016991, 10.13668999505511659984747687258, 11.11773461549232432824097086903, 12.14748416632410046598321666617

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