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Properties

Label 2-276-12.11-c1-0-26
Degree $2$
Conductor $276$
Sign $0.715 + 0.698i$
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  + (−1.04 + 0.951i)2-s + (1.08 − 1.34i)3-s + (0.189 − 1.99i)4-s + 0.289i·5-s + (0.146 + 2.44i)6-s − 1.62i·7-s + (1.69 + 2.26i)8-s + (−0.638 − 2.93i)9-s + (−0.275 − 0.303i)10-s + 0.671·11-s + (−2.48 − 2.41i)12-s + 0.807·13-s + (1.54 + 1.69i)14-s + (0.391 + 0.315i)15-s + (−3.92 − 0.752i)16-s − 3.08i·17-s + ⋯
L(s)  = 1  + (−0.739 + 0.672i)2-s + (0.627 − 0.778i)3-s + (0.0945 − 0.995i)4-s + 0.129i·5-s + (0.0599 + 0.998i)6-s − 0.613i·7-s + (0.599 + 0.800i)8-s + (−0.212 − 0.977i)9-s + (−0.0872 − 0.0959i)10-s + 0.202·11-s + (−0.715 − 0.698i)12-s + 0.223·13-s + (0.412 + 0.453i)14-s + (0.100 + 0.0813i)15-s + (−0.982 − 0.188i)16-s − 0.749i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.715 + 0.698i$
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: \((2,\ 276,\ (\ :1/2),\ 0.715 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985338 - 0.400866i\)
\(L(\frac12)\) \(\approx\) \(0.985338 - 0.400866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.951i)T \)
3 \( 1 + (-1.08 + 1.34i)T \)
23 \( 1 + T \)
good5 \( 1 - 0.289iT - 5T^{2} \)
7 \( 1 + 1.62iT - 7T^{2} \)
11 \( 1 - 0.671T + 11T^{2} \)
13 \( 1 - 0.807T + 13T^{2} \)
17 \( 1 + 3.08iT - 17T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 - 2.12iT - 31T^{2} \)
37 \( 1 - 7.38T + 37T^{2} \)
41 \( 1 - 4.70iT - 41T^{2} \)
43 \( 1 - 7.68iT - 43T^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 - 3.50iT - 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 2.00T + 61T^{2} \)
67 \( 1 - 2.36iT - 67T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + 9.56T + 73T^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + 5.78iT - 89T^{2} \)
97 \( 1 + 6.57T + 97T^{2} \)
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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

−11.59578401388107048474039571251, −10.72629758729694911157484976328, −9.500169546769124627394246091317, −8.844692840465307066966706310854, −7.76342923491123372798587848177, −7.04050594088331620903672279062, −6.25628673142542722692777459175, −4.68496832626898735303510752564, −2.81452040566163003278231247975, −1.05856626656572332906483233574, 1.99848069009532818692930825908, 3.33944305604617072706450164598, 4.37459675469194387104305879271, 5.95564816446897488243873784551, 7.63106817431175156664796741222, 8.480331029887084285334062028793, 9.152572604713511970156438197758, 10.09315567188476507055416986408, 10.79017538069469582068686465419, 11.86445865082624274627091805056

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