Properties

Label 2-28-28.27-c7-0-1
Degree $2$
Conductor $28$
Sign $-0.193 + 0.981i$
Analytic cond. $8.74678$
Root an. cond. $2.95749$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.03 + 11.2i)2-s − 29.9·3-s + (−125. + 23.3i)4-s + 473. i·5-s + (−31.1 − 337. i)6-s + (334. − 843. i)7-s + (−393. − 1.39e3i)8-s − 1.28e3·9-s + (−5.33e3 + 491. i)10-s − 1.85e3i·11-s + (3.77e3 − 700. i)12-s − 8.48e3i·13-s + (9.85e3 + 2.89e3i)14-s − 1.42e4i·15-s + (1.52e4 − 5.88e3i)16-s + 1.89e4i·17-s + ⋯
L(s)  = 1  + (0.0916 + 0.995i)2-s − 0.641·3-s + (−0.983 + 0.182i)4-s + 1.69i·5-s + (−0.0587 − 0.638i)6-s + (0.368 − 0.929i)7-s + (−0.271 − 0.962i)8-s − 0.588·9-s + (−1.68 + 0.155i)10-s − 0.420i·11-s + (0.630 − 0.117i)12-s − 1.07i·13-s + (0.959 + 0.282i)14-s − 1.08i·15-s + (0.933 − 0.358i)16-s + 0.935i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(8.74678\)
Root analytic conductor: \(2.95749\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :7/2),\ -0.193 + 0.981i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.104552 - 0.127123i\)
\(L(\frac12)\) \(\approx\) \(0.104552 - 0.127123i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 11.2i)T \)
7 \( 1 + (-334. + 843. i)T \)
good3 \( 1 + 29.9T + 2.18e3T^{2} \)
5 \( 1 - 473. iT - 7.81e4T^{2} \)
11 \( 1 + 1.85e3iT - 1.94e7T^{2} \)
13 \( 1 + 8.48e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.89e4iT - 4.10e8T^{2} \)
19 \( 1 + 2.97e4T + 8.93e8T^{2} \)
23 \( 1 - 2.04e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.30e5T + 1.72e10T^{2} \)
31 \( 1 + 2.05e5T + 2.75e10T^{2} \)
37 \( 1 + 3.08e4T + 9.49e10T^{2} \)
41 \( 1 - 6.45e5iT - 1.94e11T^{2} \)
43 \( 1 + 6.61e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.21e6T + 5.06e11T^{2} \)
53 \( 1 - 1.14e6T + 1.17e12T^{2} \)
59 \( 1 - 4.63e5T + 2.48e12T^{2} \)
61 \( 1 + 1.18e5iT - 3.14e12T^{2} \)
67 \( 1 - 1.32e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.67e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.48e5iT - 1.10e13T^{2} \)
79 \( 1 - 5.75e6iT - 1.92e13T^{2} \)
83 \( 1 + 6.75e6T + 2.71e13T^{2} \)
89 \( 1 - 1.19e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.27e7iT - 8.07e13T^{2} \)
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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

−16.70635880291431172897026068506, −15.05337148332525541927718451611, −14.47069986279374712245358814172, −13.16635133349911385699374287552, −11.14586589844374771907723664287, −10.32901649811034496943193093157, −8.085936709413037210974571996608, −6.83151314289266442831537051861, −5.71300868829287142082011054934, −3.57098529827072870361762713489, 0.084302589364504780977292089721, 1.87531368748779404101179632390, 4.57355747005335027864013729186, 5.53959619882478911747682168988, 8.592539140060140792590123144893, 9.332617062996927274955961912084, 11.32266316865101948798931578349, 12.09976899944877955321160682232, 12.98678246191190609644429784114, 14.53381864962110632892405611792

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