↑

Properties

Label 2-276-4.3-c2-0-11
Degree $2$
Conductor $276$
Sign $-0.394 - 0.918i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 + 1.10i)2-s − 1.73i·3-s + (1.57 + 3.67i)4-s − 4.68·5-s + (1.90 − 2.89i)6-s + 5.31i·7-s + (−1.41 + 7.87i)8-s − 2.99·9-s + (−7.82 − 5.15i)10-s + 18.3i·11-s + (6.36 − 2.73i)12-s − 5.17·13-s + (−5.85 + 8.87i)14-s + 8.11i·15-s + (−11.0 + 11.5i)16-s + 26.4·17-s + ⋯
L(s)  = 1  + (0.834 + 0.550i)2-s − 0.577i·3-s + (0.394 + 0.918i)4-s − 0.936·5-s + (0.317 − 0.482i)6-s + 0.759i·7-s + (−0.176 + 0.984i)8-s − 0.333·9-s + (−0.782 − 0.515i)10-s + 1.67i·11-s + (0.530 − 0.227i)12-s − 0.397·13-s + (−0.417 + 0.634i)14-s + 0.540i·15-s + (−0.688 + 0.724i)16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.394 - 0.918i$
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: \((2,\ 276,\ (\ :1),\ -0.394 - 0.918i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03426 + 1.56943i\)
\(L(\frac12)\) \(\approx\) \(1.03426 + 1.56943i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.66 - 1.10i)T \)
3 \( 1 + 1.73iT \)
23 \( 1 - 4.79iT \)
good5 \( 1 + 4.68T + 25T^{2} \)
7 \( 1 - 5.31iT - 49T^{2} \)
11 \( 1 - 18.3iT - 121T^{2} \)
13 \( 1 + 5.17T + 169T^{2} \)
17 \( 1 - 26.4T + 289T^{2} \)
19 \( 1 - 3.79iT - 361T^{2} \)
29 \( 1 - 10.6T + 841T^{2} \)
31 \( 1 + 17.0iT - 961T^{2} \)
37 \( 1 + 33.2T + 1.36e3T^{2} \)
41 \( 1 - 17.0T + 1.68e3T^{2} \)
43 \( 1 + 67.4iT - 1.84e3T^{2} \)
47 \( 1 - 19.2iT - 2.20e3T^{2} \)
53 \( 1 - 8.27T + 2.80e3T^{2} \)
59 \( 1 - 71.6iT - 3.48e3T^{2} \)
61 \( 1 - 66.2T + 3.72e3T^{2} \)
67 \( 1 - 25.1iT - 4.48e3T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 + 67.2T + 5.32e3T^{2} \)
79 \( 1 - 11.2iT - 6.24e3T^{2} \)
83 \( 1 - 72.0iT - 6.88e3T^{2} \)
89 \( 1 - 143.T + 7.92e3T^{2} \)
97 \( 1 - 121.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.22656077160015923095952594098, −11.65069363939108547544041211040, −10.09825845480816293379551472359, −8.737326903242282823524937395371, −7.59814754608365331145162842748, −7.28596004668050548025009935199, −5.89571056765496490582122775092, −4.88023811131931775531509911540, −3.64749769975115664836207389461, −2.22549358179299751645397646689, 0.73255032416071685048338058450, 3.15874243241035640414879790789, 3.78931037155721436445592310823, 4.96398420013183453628443907044, 6.04578496528609258818613111733, 7.39610799781153619798534024279, 8.486362755249199073664358289152, 9.852295302881220669561592593133, 10.64625733517778068926535253428, 11.41553774869629062185392904278

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