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Properties

Label 2-276-12.11-c1-0-32
Degree $2$
Conductor $276$
Sign $0.120 + 0.992i$
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  + (0.565 − 1.29i)2-s + (1.40 + 1.01i)3-s + (−1.35 − 1.46i)4-s − 0.662i·5-s + (2.11 − 1.24i)6-s − 4.60i·7-s + (−2.67 + 0.933i)8-s + (0.932 + 2.85i)9-s + (−0.859 − 0.375i)10-s + 1.50·11-s + (−0.416 − 3.43i)12-s + 1.18·13-s + (−5.96 − 2.60i)14-s + (0.673 − 0.929i)15-s + (−0.301 + 3.98i)16-s + 1.95i·17-s + ⋯
L(s)  = 1  + (0.400 − 0.916i)2-s + (0.809 + 0.586i)3-s + (−0.679 − 0.733i)4-s − 0.296i·5-s + (0.861 − 0.507i)6-s − 1.73i·7-s + (−0.944 + 0.329i)8-s + (0.310 + 0.950i)9-s + (−0.271 − 0.118i)10-s + 0.452·11-s + (−0.120 − 0.992i)12-s + 0.328·13-s + (−1.59 − 0.695i)14-s + (0.174 − 0.240i)15-s + (−0.0752 + 0.997i)16-s + 0.473i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.120 + 0.992i$
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: \((2,\ 276,\ (\ :1/2),\ 0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37523 - 1.21882i\)
\(L(\frac12)\) \(\approx\) \(1.37523 - 1.21882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.565 + 1.29i)T \)
3 \( 1 + (-1.40 - 1.01i)T \)
23 \( 1 + T \)
good5 \( 1 + 0.662iT - 5T^{2} \)
7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
13 \( 1 - 1.18T + 13T^{2} \)
17 \( 1 - 1.95iT - 17T^{2} \)
19 \( 1 + 0.0109iT - 19T^{2} \)
29 \( 1 - 8.65iT - 29T^{2} \)
31 \( 1 + 6.06iT - 31T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
41 \( 1 - 7.37iT - 41T^{2} \)
43 \( 1 - 4.67iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 0.825iT - 53T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 0.812iT - 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 9.43iT - 79T^{2} \)
83 \( 1 - 1.34T + 83T^{2} \)
89 \( 1 + 6.30iT - 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−11.44350781122386515320008283191, −10.61266326121410757817521627932, −10.02261156288163300446618989760, −9.039524664126477130661604595246, −8.065365311454567879802041645129, −6.72453146007455523885868994686, −4.98021719677431372250291584376, −4.08484786086872483039659555991, −3.30184250805725569581587923961, −1.42672955132390386471448767754, 2.42701218574792894241598234067, 3.58478783947041897228023138881, 5.22320527754243992230274862341, 6.28538112450589249713410741362, 7.07788120654300456433247498186, 8.380533161765351727670286979638, 8.784291433988117004220582798312, 9.739644277831152576728332774346, 11.68438056703293800419240636163, 12.27422567324991405351575034201

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