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Properties

Label 2-276-12.11-c1-0-27
Degree $2$
Conductor $276$
Sign $0.999 - 0.0278i$
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  + (1.33 − 0.466i)2-s + (1.38 + 1.04i)3-s + (1.56 − 1.24i)4-s + 0.588i·5-s + (2.33 + 0.744i)6-s + 0.538i·7-s + (1.50 − 2.39i)8-s + (0.830 + 2.88i)9-s + (0.274 + 0.785i)10-s − 5.85·11-s + (3.46 − 0.0964i)12-s − 1.97·13-s + (0.251 + 0.718i)14-s + (−0.613 + 0.814i)15-s + (0.892 − 3.89i)16-s − 7.26i·17-s + ⋯
L(s)  = 1  + (0.943 − 0.330i)2-s + (0.799 + 0.601i)3-s + (0.781 − 0.623i)4-s + 0.263i·5-s + (0.952 + 0.303i)6-s + 0.203i·7-s + (0.532 − 0.846i)8-s + (0.276 + 0.960i)9-s + (0.0869 + 0.248i)10-s − 1.76·11-s + (0.999 − 0.0278i)12-s − 0.548·13-s + (0.0671 + 0.191i)14-s + (−0.158 + 0.210i)15-s + (0.223 − 0.974i)16-s − 1.76i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.999 - 0.0278i$
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: \((2,\ 276,\ (\ :1/2),\ 0.999 - 0.0278i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55805 + 0.0356129i\)
\(L(\frac12)\) \(\approx\) \(2.55805 + 0.0356129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.466i)T \)
3 \( 1 + (-1.38 - 1.04i)T \)
23 \( 1 - T \)
good5 \( 1 - 0.588iT - 5T^{2} \)
7 \( 1 - 0.538iT - 7T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
13 \( 1 + 1.97T + 13T^{2} \)
17 \( 1 + 7.26iT - 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
29 \( 1 + 4.53iT - 29T^{2} \)
31 \( 1 - 9.26iT - 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 - 1.51iT - 41T^{2} \)
43 \( 1 + 4.18iT - 43T^{2} \)
47 \( 1 + 8.82T + 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 + 2.25T + 61T^{2} \)
67 \( 1 + 7.43iT - 67T^{2} \)
71 \( 1 - 3.93T + 71T^{2} \)
73 \( 1 - 0.236T + 73T^{2} \)
79 \( 1 - 7.96iT - 79T^{2} \)
83 \( 1 + 0.731T + 83T^{2} \)
89 \( 1 - 3.18iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−11.99936784333983679980446755643, −10.80638598573160806352434897988, −10.25210318969577679242375428610, −9.260235654617655892364297661779, −7.87475074597616269621294177929, −7.01636051136229510148440400059, −5.30493543315647444893018100844, −4.75724252374874482423048674342, −3.12716220258022834685408845186, −2.48798912658700336077763222797, 2.14846448789127861442409646131, 3.30790847886582493379241582772, 4.65177242834360296360089433097, 5.83590593507121456140582140174, 7.01977989411790078697358085978, 7.890841120959911007683216060035, 8.551313692159136801232931230098, 10.10032256895236742422833099796, 11.08999211683701193253348958035, 12.42629478814438678469068143035

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