↑

Properties

Label 2-276-12.11-c1-0-11
Degree $2$
Conductor $276$
Sign $-0.903 - 0.429i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.456 + 1.33i)2-s + (0.784 + 1.54i)3-s + (−1.58 + 1.22i)4-s + 1.42i·5-s + (−1.70 + 1.75i)6-s − 1.12i·7-s + (−2.35 − 1.56i)8-s + (−1.77 + 2.42i)9-s + (−1.90 + 0.649i)10-s − 1.68·11-s + (−3.12 − 1.48i)12-s + 5.14·13-s + (1.50 − 0.513i)14-s + (−2.19 + 1.11i)15-s + (1.01 − 3.86i)16-s + 4.65i·17-s + ⋯
L(s)  = 1  + (0.322 + 0.946i)2-s + (0.452 + 0.891i)3-s + (−0.791 + 0.610i)4-s + 0.636i·5-s + (−0.697 + 0.716i)6-s − 0.425i·7-s + (−0.833 − 0.552i)8-s + (−0.590 + 0.807i)9-s + (−0.602 + 0.205i)10-s − 0.507·11-s + (−0.903 − 0.429i)12-s + 1.42·13-s + (0.402 − 0.137i)14-s + (−0.567 + 0.288i)15-s + (0.253 − 0.967i)16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.903 - 0.429i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ -0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332031 + 1.47128i\)
\(L(\frac12)\) \(\approx\) \(0.332031 + 1.47128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.456 - 1.33i)T \)
3 \( 1 + (-0.784 - 1.54i)T \)
23 \( 1 - T \)
good5 \( 1 - 1.42iT - 5T^{2} \)
7 \( 1 + 1.12iT - 7T^{2} \)
11 \( 1 + 1.68T + 11T^{2} \)
13 \( 1 - 5.14T + 13T^{2} \)
17 \( 1 - 4.65iT - 17T^{2} \)
19 \( 1 + 5.86iT - 19T^{2} \)
29 \( 1 - 3.67iT - 29T^{2} \)
31 \( 1 - 3.79iT - 31T^{2} \)
37 \( 1 + 2.52T + 37T^{2} \)
41 \( 1 + 8.23iT - 41T^{2} \)
43 \( 1 - 3.72iT - 43T^{2} \)
47 \( 1 - 7.62T + 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + 3.98T + 59T^{2} \)
61 \( 1 + 7.14T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 + 3.55T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 5.27iT - 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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

−12.59160594779578627723783646075, −10.91625615680669746790203431180, −10.56331852362145123256496323738, −9.110250651599318723483009568576, −8.506101392953300059466822132504, −7.38673198622285837279952498238, −6.33460157521210903508372287213, −5.16778514105703081715043116418, −4.00924063811188230227536804015, −3.07636280201761253392868984561, 1.12398455331080918405575204036, 2.54489885727019502804143570611, 3.80786460576964790088184820872, 5.32310696033698963905360964733, 6.25225234331437271372173991304, 7.891625037309704357315182556191, 8.710027004353670164669201386854, 9.467230885273615173331933464547, 10.74324149615978363678462274763, 11.78725158564962132850121954422

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