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Properties

Label 2-276-4.3-c2-0-13
Degree $2$
Conductor $276$
Sign $-0.769 - 0.638i$
Analytic cond. $7.52045$
Root an. cond. $2.74234$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.678 + 1.88i)2-s + 1.73i·3-s + (−3.07 − 2.55i)4-s + 4.99·5-s + (−3.25 − 1.17i)6-s − 2.08i·7-s + (6.89 − 4.05i)8-s − 2.99·9-s + (−3.38 + 9.38i)10-s + 19.2i·11-s + (4.42 − 5.33i)12-s − 0.674·13-s + (3.92 + 1.41i)14-s + 8.64i·15-s + (2.95 + 15.7i)16-s + 14.9·17-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)2-s + 0.577i·3-s + (−0.769 − 0.638i)4-s + 0.998·5-s + (−0.543 − 0.195i)6-s − 0.298i·7-s + (0.861 − 0.507i)8-s − 0.333·9-s + (−0.338 + 0.938i)10-s + 1.75i·11-s + (0.368 − 0.444i)12-s − 0.0519·13-s + (0.280 + 0.101i)14-s + 0.576i·15-s + (0.184 + 0.982i)16-s + 0.882·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(7.52045\)
Root analytic conductor: \(2.74234\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.460133 + 1.27538i\)
\(L(\frac12)\) \(\approx\) \(0.460133 + 1.27538i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.678 - 1.88i)T \)
3 \( 1 - 1.73iT \)
23 \( 1 - 4.79iT \)
good5 \( 1 - 4.99T + 25T^{2} \)
7 \( 1 + 2.08iT - 49T^{2} \)
11 \( 1 - 19.2iT - 121T^{2} \)
13 \( 1 + 0.674T + 169T^{2} \)
17 \( 1 - 14.9T + 289T^{2} \)
19 \( 1 - 26.4iT - 361T^{2} \)
29 \( 1 + 17.9T + 841T^{2} \)
31 \( 1 + 5.50iT - 961T^{2} \)
37 \( 1 + 8.99T + 1.36e3T^{2} \)
41 \( 1 - 9.20T + 1.68e3T^{2} \)
43 \( 1 - 67.6iT - 1.84e3T^{2} \)
47 \( 1 + 77.8iT - 2.20e3T^{2} \)
53 \( 1 + 10.3T + 2.80e3T^{2} \)
59 \( 1 - 13.2iT - 3.48e3T^{2} \)
61 \( 1 + 17.0T + 3.72e3T^{2} \)
67 \( 1 - 39.0iT - 4.48e3T^{2} \)
71 \( 1 + 42.7iT - 5.04e3T^{2} \)
73 \( 1 - 98.4T + 5.32e3T^{2} \)
79 \( 1 - 34.1iT - 6.24e3T^{2} \)
83 \( 1 - 91.0iT - 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 - 16.5T + 9.40e3T^{2} \)
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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

−12.15870717341918952278022843063, −10.53120962734897832203540913896, −9.830660210250667717999060261575, −9.482191782543703254964640912724, −8.073427952960714596181496223176, −7.16430680744834690612666818514, −5.98098476381455794218094760917, −5.14218453446345799024671996032, −3.98335285855721247472862561204, −1.75398928723534530949130793962, 0.808695540504306000606906915327, 2.28993529872182173288446186706, 3.37395180613659305271902672625, 5.22497753996155658886305107006, 6.15138182968058290464003497333, 7.61031488392515191859645949727, 8.723519542260457978591982152309, 9.309558607558672349965760804762, 10.50017513196055985030201976291, 11.25114182196447144729211040122

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