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Properties

Label 2-276-4.3-c2-0-29
Degree $2$
Conductor $276$
Sign $-0.150 + 0.988i$
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  + (−1.51 + 1.30i)2-s + 1.73i·3-s + (0.600 − 3.95i)4-s − 0.0471·5-s + (−2.25 − 2.62i)6-s − 2.88i·7-s + (4.24 + 6.78i)8-s − 2.99·9-s + (0.0714 − 0.0614i)10-s + 0.613i·11-s + (6.84 + 1.03i)12-s − 19.6·13-s + (3.76 + 4.38i)14-s − 0.0816i·15-s + (−15.2 − 4.74i)16-s − 16.6·17-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)2-s + 0.577i·3-s + (0.150 − 0.988i)4-s − 0.00942·5-s + (−0.376 − 0.437i)6-s − 0.412i·7-s + (0.530 + 0.847i)8-s − 0.333·9-s + (0.00714 − 0.00614i)10-s + 0.0557i·11-s + (0.570 + 0.0866i)12-s − 1.50·13-s + (0.268 + 0.312i)14-s − 0.00544i·15-s + (−0.954 − 0.296i)16-s − 0.980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.150 + 0.988i$
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: \((2,\ 276,\ (\ :1),\ -0.150 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.159113 - 0.185089i\)
\(L(\frac12)\) \(\approx\) \(0.159113 - 0.185089i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 - 1.30i)T \)
3 \( 1 - 1.73iT \)
23 \( 1 + 4.79iT \)
good5 \( 1 + 0.0471T + 25T^{2} \)
7 \( 1 + 2.88iT - 49T^{2} \)
11 \( 1 - 0.613iT - 121T^{2} \)
13 \( 1 + 19.6T + 169T^{2} \)
17 \( 1 + 16.6T + 289T^{2} \)
19 \( 1 + 20.5iT - 361T^{2} \)
29 \( 1 - 22.9T + 841T^{2} \)
31 \( 1 + 5.53iT - 961T^{2} \)
37 \( 1 + 28.4T + 1.36e3T^{2} \)
41 \( 1 - 18.1T + 1.68e3T^{2} \)
43 \( 1 + 58.0iT - 1.84e3T^{2} \)
47 \( 1 + 51.1iT - 2.20e3T^{2} \)
53 \( 1 + 66.3T + 2.80e3T^{2} \)
59 \( 1 - 12.3iT - 3.48e3T^{2} \)
61 \( 1 + 74.6T + 3.72e3T^{2} \)
67 \( 1 + 63.3iT - 4.48e3T^{2} \)
71 \( 1 - 94.5iT - 5.04e3T^{2} \)
73 \( 1 + 40.7T + 5.32e3T^{2} \)
79 \( 1 - 122. iT - 6.24e3T^{2} \)
83 \( 1 - 27.6iT - 6.88e3T^{2} \)
89 \( 1 - 17.1T + 7.92e3T^{2} \)
97 \( 1 + 117.T + 9.40e3T^{2} \)
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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

−11.07141673607457879368470879809, −10.24952838227073763174704884601, −9.478661602658804930512911251852, −8.621556513697776694553863703849, −7.45828382402474303254300635555, −6.67226367541099213346788821713, −5.29391496719183921395607941727, −4.38507908270793084252834152127, −2.37553516679975338715527600775, −0.14719125063723135110022259247, 1.79740342194286661512946342816, 2.92426427853652939352447852493, 4.54368178021776095857456317708, 6.15740439686991990790298634008, 7.34112800886590855731563044446, 8.082604033680040989856099532445, 9.152450375869830007478672628195, 9.965034718426136846866091383059, 11.00900340287042888915251382331, 12.03926025763321776190513353332

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