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Properties

Label 2-28-28.3-c3-0-8
Degree $2$
Conductor $28$
Sign $0.770 + 0.637i$
Analytic cond. $1.65205$
Root an. cond. $1.28532$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.11i)2-s + (1.67 − 2.89i)3-s + (5.51 − 5.79i)4-s + (−15.9 + 9.21i)5-s + (1.11 − 9.38i)6-s + (15.4 + 10.1i)7-s + (7.87 − 21.2i)8-s + (7.91 + 13.7i)9-s + (−31.2 + 41.7i)10-s + (−35.6 − 20.5i)11-s + (−7.55 − 25.6i)12-s + 24.8i·13-s + (51.5 + 9.15i)14-s + 61.5i·15-s + (−3.17 − 63.9i)16-s + (−41.3 − 23.8i)17-s + ⋯
L(s)  = 1  + (0.919 − 0.394i)2-s + (0.321 − 0.557i)3-s + (0.689 − 0.724i)4-s + (−1.42 + 0.824i)5-s + (0.0760 − 0.638i)6-s + (0.836 + 0.548i)7-s + (0.348 − 0.937i)8-s + (0.293 + 0.507i)9-s + (−0.987 + 1.31i)10-s + (−0.976 − 0.563i)11-s + (−0.181 − 0.617i)12-s + 0.530i·13-s + (0.984 + 0.174i)14-s + 1.06i·15-s + (−0.0495 − 0.998i)16-s + (−0.590 − 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68700 - 0.607215i\)
\(L(\frac12)\) \(\approx\) \(1.68700 - 0.607215i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.59 + 1.11i)T \)
7 \( 1 + (-15.4 - 10.1i)T \)
good3 \( 1 + (-1.67 + 2.89i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (15.9 - 9.21i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (35.6 + 20.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 24.8iT - 2.19e3T^{2} \)
17 \( 1 + (41.3 + 23.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (30.9 + 53.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-64.3 + 37.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 28.8T + 2.43e4T^{2} \)
31 \( 1 + (11.5 - 20.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-51.8 - 89.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 96.1iT - 6.89e4T^{2} \)
43 \( 1 - 195. iT - 7.95e4T^{2} \)
47 \( 1 + (89.8 + 155. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (218. - 378. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-286. + 496. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-368. + 212. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (728. + 420. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.17e3iT - 3.57e5T^{2} \)
73 \( 1 + (-716. - 413. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-282. + 162. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 507.T + 5.71e5T^{2} \)
89 \( 1 + (176. - 102. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.11e3iT - 9.12e5T^{2} \)
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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

−16.00367541041139239947294267481, −15.19732234568096460426080420717, −14.12140424849087530640071222874, −12.86980001318652994040633592496, −11.50078107281219830159454152483, −10.85389115256676047163837532443, −8.156378480858217039496292944146, −6.94900812924394475145367850722, −4.69003770895098075927118344608, −2.67568489860983307880233048268, 3.87225403526749509556141788980, 4.87559756416934249500943738162, 7.44534898172188509202955805617, 8.432401459314354281395232469016, 10.74057390492284229198038175733, 12.09651379320132168414952089805, 13.08800170439836083810633781300, 14.84082455209422376167584365387, 15.42032056865775769240201812954, 16.36787250597633340550728665759

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