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Properties

Label 2-28-4.3-c8-0-21
Degree $2$
Conductor $28$
Sign $0.143 + 0.989i$
Analytic cond. $11.4066$
Root an. cond. $3.37736$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 − 1.15i)2-s − 111. i·3-s + (253. − 36.8i)4-s + 600.·5-s + (−128. − 1.77e3i)6-s + 907. i·7-s + (4.00e3 − 880. i)8-s − 5.83e3·9-s + (9.58e3 − 692. i)10-s − 1.70e4i·11-s + (−4.10e3 − 2.82e4i)12-s − 4.63e4·13-s + (1.04e3 + 1.44e4i)14-s − 6.68e4i·15-s + (6.28e4 − 1.86e4i)16-s + 6.80e4·17-s + ⋯
L(s)  = 1  + (0.997 − 0.0721i)2-s − 1.37i·3-s + (0.989 − 0.143i)4-s + 0.960·5-s + (−0.0991 − 1.37i)6-s + 0.377i·7-s + (0.976 − 0.214i)8-s − 0.889·9-s + (0.958 − 0.0692i)10-s − 1.16i·11-s + (−0.197 − 1.36i)12-s − 1.62·13-s + (0.0272 + 0.376i)14-s − 1.32i·15-s + (0.958 − 0.284i)16-s + 0.814·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(11.4066\)
Root analytic conductor: \(3.37736\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :4),\ 0.143 + 0.989i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.76823 - 2.39491i\)
\(L(\frac12)\) \(\approx\) \(2.76823 - 2.39491i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-15.9 + 1.15i)T \)
7 \( 1 - 907. iT \)
good3 \( 1 + 111. iT - 6.56e3T^{2} \)
5 \( 1 - 600.T + 3.90e5T^{2} \)
11 \( 1 + 1.70e4iT - 2.14e8T^{2} \)
13 \( 1 + 4.63e4T + 8.15e8T^{2} \)
17 \( 1 - 6.80e4T + 6.97e9T^{2} \)
19 \( 1 - 1.77e5iT - 1.69e10T^{2} \)
23 \( 1 - 6.96e3iT - 7.83e10T^{2} \)
29 \( 1 - 8.21e5T + 5.00e11T^{2} \)
31 \( 1 - 1.40e6iT - 8.52e11T^{2} \)
37 \( 1 - 5.68e5T + 3.51e12T^{2} \)
41 \( 1 + 1.36e6T + 7.98e12T^{2} \)
43 \( 1 - 2.05e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.75e6iT - 2.38e13T^{2} \)
53 \( 1 - 8.25e6T + 6.22e13T^{2} \)
59 \( 1 + 1.52e7iT - 1.46e14T^{2} \)
61 \( 1 - 9.22e5T + 1.91e14T^{2} \)
67 \( 1 - 1.72e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.82e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.45e7T + 8.06e14T^{2} \)
79 \( 1 - 6.21e7iT - 1.51e15T^{2} \)
83 \( 1 + 4.44e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.35e7T + 3.93e15T^{2} \)
97 \( 1 + 1.13e8T + 7.83e15T^{2} \)
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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

−14.45358602175658088533559067388, −13.85667132170788885203658692077, −12.63403712917424575346252368933, −11.95193138425331840010323390671, −10.12216276655014570006825764257, −7.902455373705010581195487008856, −6.48339007138040798211238593055, −5.45161114378916739141489568370, −2.76425074513683922566015890608, −1.42895213700853421716821252567, 2.47456037088480395120105903786, 4.36267338353830239819070668824, 5.29775930750511248543187855645, 7.15529119604689099939463994210, 9.687486296775442454361084654303, 10.31055592369219969396951741225, 11.96926038987022440007231980141, 13.40116891689427931700474178830, 14.66597215090565250282268385359, 15.31887154164003774103330622481

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