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Properties

Label 2-28-28.11-c2-0-3
Degree $2$
Conductor $28$
Sign $0.775 + 0.631i$
Analytic cond. $0.762944$
Root an. cond. $0.873467$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.371 − 1.96i)2-s + (3.95 + 2.28i)3-s + (−3.72 + 1.46i)4-s + (−2.62 − 4.54i)5-s + (3.01 − 8.60i)6-s + (−5.86 + 3.81i)7-s + (4.25 + 6.77i)8-s + (5.90 + 10.2i)9-s + (−7.96 + 6.85i)10-s + (1.91 + 1.10i)11-s + (−18.0 − 2.72i)12-s − 3.29·13-s + (9.68 + 10.1i)14-s − 23.9i·15-s + (11.7 − 10.8i)16-s + (6.69 − 11.6i)17-s + ⋯
L(s)  = 1  + (−0.185 − 0.982i)2-s + (1.31 + 0.760i)3-s + (−0.930 + 0.365i)4-s + (−0.525 − 0.909i)5-s + (0.502 − 1.43i)6-s + (−0.838 + 0.545i)7-s + (0.531 + 0.846i)8-s + (0.655 + 1.13i)9-s + (−0.796 + 0.685i)10-s + (0.174 + 0.100i)11-s + (−1.50 − 0.227i)12-s − 0.253·13-s + (0.691 + 0.722i)14-s − 1.59i·15-s + (0.733 − 0.679i)16-s + (0.394 − 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.775 + 0.631i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.975017 - 0.346993i\)
\(L(\frac12)\) \(\approx\) \(0.975017 - 0.346993i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.371 + 1.96i)T \)
7 \( 1 + (5.86 - 3.81i)T \)
good3 \( 1 + (-3.95 - 2.28i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.62 + 4.54i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.91 - 1.10i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 3.29T + 169T^{2} \)
17 \( 1 + (-6.69 + 11.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (6.72 - 3.88i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.66 + 2.11i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 39.4T + 841T^{2} \)
31 \( 1 + (17.2 + 9.93i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.8 - 22.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 - 78.2iT - 1.84e3T^{2} \)
47 \( 1 + (-18.3 + 10.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (24.0 - 41.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (66.7 + 38.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.7 - 41.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-45.2 - 26.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 90.1iT - 5.04e3T^{2} \)
73 \( 1 + (20.7 - 36.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (114. - 65.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 11.8iT - 6.88e3T^{2} \)
89 \( 1 + (15.5 + 26.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 140.T + 9.40e3T^{2} \)
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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

−16.72222311899002013129719088214, −15.65005444517762100258809095899, −14.32034707608399564587992604774, −13.02374441014509763368355063930, −11.96152418597254759139598398407, −10.01719122102972599794044540454, −9.127144568026715598516303070599, −8.203225191696049177973478588184, −4.55052845421843727426311663743, −3.02206622387860335603793497967, 3.50884044642943070170020843447, 6.66854990185344846798080280158, 7.51965406728441988532918071456, 8.787053394572759153692466582833, 10.29566082531867212504194449745, 12.72178778453519010915664074196, 13.84271892539544143400194332484, 14.65472870090998384155929128806, 15.65125908707645455594479219809, 17.13499172591238997951984575164

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