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Properties

Label 2-276-12.11-c1-0-37
Degree $2$
Conductor $276$
Sign $-0.911 + 0.410i$
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  + (−1.27 − 0.610i)2-s + (0.436 − 1.67i)3-s + (1.25 + 1.55i)4-s − 3.87i·5-s + (−1.57 + 1.87i)6-s + 0.468i·7-s + (−0.649 − 2.75i)8-s + (−2.61 − 1.46i)9-s + (−2.36 + 4.94i)10-s − 0.650·11-s + (3.15 − 1.42i)12-s + 3.42·13-s + (0.285 − 0.597i)14-s + (−6.49 − 1.68i)15-s + (−0.852 + 3.90i)16-s + 6.36i·17-s + ⋯
L(s)  = 1  + (−0.902 − 0.431i)2-s + (0.251 − 0.967i)3-s + (0.627 + 0.778i)4-s − 1.73i·5-s + (−0.644 + 0.764i)6-s + 0.176i·7-s + (−0.229 − 0.973i)8-s + (−0.873 − 0.487i)9-s + (−0.748 + 1.56i)10-s − 0.196·11-s + (0.911 − 0.410i)12-s + 0.949·13-s + (0.0764 − 0.159i)14-s + (−1.67 − 0.436i)15-s + (−0.213 + 0.977i)16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.911 + 0.410i$
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: \((2,\ 276,\ (\ :1/2),\ -0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175596 - 0.816813i\)
\(L(\frac12)\) \(\approx\) \(0.175596 - 0.816813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.610i)T \)
3 \( 1 + (-0.436 + 1.67i)T \)
23 \( 1 + T \)
good5 \( 1 + 3.87iT - 5T^{2} \)
7 \( 1 - 0.468iT - 7T^{2} \)
11 \( 1 + 0.650T + 11T^{2} \)
13 \( 1 - 3.42T + 13T^{2} \)
17 \( 1 - 6.36iT - 17T^{2} \)
19 \( 1 + 7.16iT - 19T^{2} \)
29 \( 1 + 0.0999iT - 29T^{2} \)
31 \( 1 - 3.26iT - 31T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 + 6.03iT - 41T^{2} \)
43 \( 1 + 6.56iT - 43T^{2} \)
47 \( 1 - 0.906T + 47T^{2} \)
53 \( 1 - 6.73iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 6.94iT - 67T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 - 2.07T + 73T^{2} \)
79 \( 1 + 5.03iT - 79T^{2} \)
83 \( 1 + 4.30T + 83T^{2} \)
89 \( 1 - 1.13iT - 89T^{2} \)
97 \( 1 - 9.48T + 97T^{2} \)
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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

−11.63428242280882691850466383970, −10.55729908468747310894455353780, −9.047920495383081979031858309421, −8.728414678818661077963212460668, −8.004470203606110185595144690586, −6.76226691037961892959436677567, −5.53566915910329838386811001756, −3.83877845973955330983098022760, −2.05276781274914256328828184761, −0.846741613543932229046052797925, 2.52629314649094420316960221617, 3.68357236199077385854908980445, 5.51590132124929552212484057247, 6.49904031300038714307164093133, 7.54051888151272836979791741106, 8.431824584245611904670251176031, 9.782144016852941062504359491040, 10.13162549479683716143644248818, 11.11132059178502936012289658136, 11.56359012291313730733148308318

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