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Properties

Label 2-28-28.11-c2-0-1
Degree $2$
Conductor $28$
Sign $0.371 - 0.928i$
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  + (−0.104 + 1.99i)2-s + (1.63 + 0.941i)3-s + (−3.97 − 0.418i)4-s + (−1.12 − 1.95i)5-s + (−2.05 + 3.15i)6-s + (6.84 + 1.47i)7-s + (1.25 − 7.90i)8-s + (−2.72 − 4.72i)9-s + (4.01 − 2.04i)10-s + (−8.47 − 4.89i)11-s + (−6.09 − 4.42i)12-s + 7.96·13-s + (−3.66 + 13.5i)14-s − 4.24i·15-s + (15.6 + 3.33i)16-s + (−13.1 + 22.8i)17-s + ⋯
L(s)  = 1  + (−0.0523 + 0.998i)2-s + (0.543 + 0.313i)3-s + (−0.994 − 0.104i)4-s + (−0.225 − 0.390i)5-s + (−0.341 + 0.526i)6-s + (0.977 + 0.210i)7-s + (0.156 − 0.987i)8-s + (−0.303 − 0.525i)9-s + (0.401 − 0.204i)10-s + (−0.770 − 0.444i)11-s + (−0.507 − 0.368i)12-s + 0.612·13-s + (−0.261 + 0.965i)14-s − 0.282i·15-s + (0.978 + 0.208i)16-s + (−0.775 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.371 - 0.928i$
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: \((2,\ 28,\ (\ :1),\ 0.371 - 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.820761 + 0.555616i\)
\(L(\frac12)\) \(\approx\) \(0.820761 + 0.555616i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 - 1.99i)T \)
7 \( 1 + (-6.84 - 1.47i)T \)
good3 \( 1 + (-1.63 - 0.941i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.12 + 1.95i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.47 + 4.89i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 7.96T + 169T^{2} \)
17 \( 1 + (13.1 - 22.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (21.1 - 12.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (3.55 - 2.05i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 12.3T + 841T^{2} \)
31 \( 1 + (-44.2 - 25.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (16.7 + 29.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 31.2T + 1.68e3T^{2} \)
43 \( 1 - 21.4iT - 1.84e3T^{2} \)
47 \( 1 + (-39.4 + 22.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (7.90 - 13.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (54.5 + 31.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 - 32.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (28.1 + 16.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.1iT - 5.04e3T^{2} \)
73 \( 1 + (9.53 - 16.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (4.94 - 2.85i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 37.5iT - 6.88e3T^{2} \)
89 \( 1 + (-4.70 - 8.14i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 64.3T + 9.40e3T^{2} \)
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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

−17.18344871184113528810806360242, −15.80203975487792141134087321639, −14.98321288676584533106668498679, −13.96885586629844487690351930884, −12.55981544780182301770979359055, −10.60994968694136845217778004151, −8.693626042336914714185248909934, −8.229531974544539084227539465612, −6.03988212614111218277664787937, −4.24505690717350605612763212825, 2.47432806685604655148179334711, 4.74181392359231922408763513384, 7.64922698059513795283929214900, 8.827196501092492065928010672257, 10.65420551493516948190959097240, 11.47436452803946553595540478531, 13.17773296421602769090044624274, 13.97387834403070315757268024774, 15.26126660417313861901396592747, 17.23222587860552906337631480256

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