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Properties

Label 2-276-12.11-c1-0-14
Degree $2$
Conductor $276$
Sign $0.989 - 0.141i$
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  + (−0.133 − 1.40i)2-s + (1.73 + 0.0815i)3-s + (−1.96 + 0.376i)4-s + 3.66i·5-s + (−0.116 − 2.44i)6-s + 2.47i·7-s + (0.793 + 2.71i)8-s + (2.98 + 0.282i)9-s + (5.16 − 0.490i)10-s − 4.16·11-s + (−3.42 + 0.491i)12-s + 3.10·13-s + (3.48 − 0.331i)14-s + (−0.299 + 6.34i)15-s + (3.71 − 1.47i)16-s + 0.828i·17-s + ⋯
L(s)  = 1  + (−0.0945 − 0.995i)2-s + (0.998 + 0.0470i)3-s + (−0.982 + 0.188i)4-s + 1.64i·5-s + (−0.0476 − 0.998i)6-s + 0.935i·7-s + (0.280 + 0.959i)8-s + (0.995 + 0.0940i)9-s + (1.63 − 0.155i)10-s − 1.25·11-s + (−0.989 + 0.141i)12-s + 0.861·13-s + (0.931 − 0.0885i)14-s + (−0.0772 + 1.63i)15-s + (0.929 − 0.369i)16-s + 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.989 - 0.141i$
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: \((2,\ 276,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46750 + 0.104649i\)
\(L(\frac12)\) \(\approx\) \(1.46750 + 0.104649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.133 + 1.40i)T \)
3 \( 1 + (-1.73 - 0.0815i)T \)
23 \( 1 + T \)
good5 \( 1 - 3.66iT - 5T^{2} \)
7 \( 1 - 2.47iT - 7T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 - 3.10T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + 7.27iT - 19T^{2} \)
29 \( 1 + 2.16iT - 29T^{2} \)
31 \( 1 + 6.69iT - 31T^{2} \)
37 \( 1 - 9.22T + 37T^{2} \)
41 \( 1 + 1.93iT - 41T^{2} \)
43 \( 1 - 1.56iT - 43T^{2} \)
47 \( 1 - 5.50T + 47T^{2} \)
53 \( 1 - 5.10iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 0.476T + 61T^{2} \)
67 \( 1 - 9.83iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.39T + 73T^{2} \)
79 \( 1 + 1.28iT - 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 6.94iT - 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−11.64059563698710169804359390410, −10.89793344848816938374084962371, −10.13865736520848402047486253948, −9.199187850219327551215280231626, −8.242043473415909876315248491239, −7.30013206432114519816846399483, −5.81203347043339490826147649410, −4.15419256488626341099771785852, −2.76664538620055330958831140662, −2.50555699529466706823279012035, 1.20787654213889985033125335909, 3.75253423009696561573507275150, 4.66466713024516291190293255958, 5.76991509730317584395834378672, 7.31546576922640647806393706804, 8.151366975738603456840971630070, 8.589946865324442394311824134377, 9.688140998896076080997062493315, 10.44770799582739536777012869905, 12.43372207194448585262009453993

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