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Properties

Label 2-28-4.3-c2-0-5
Degree $2$
Conductor $28$
Sign $0.163 + 0.986i$
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  + (−0.163 − 1.99i)2-s − 1.56i·3-s + (−3.94 + 0.652i)4-s + 3.43·5-s + (−3.11 + 0.255i)6-s + 2.64i·7-s + (1.94 + 7.75i)8-s + 6.56·9-s + (−0.562 − 6.85i)10-s + 8.48i·11-s + (1.01 + 6.15i)12-s − 18.5·13-s + (5.27 − 0.433i)14-s − 5.36i·15-s + (15.1 − 5.14i)16-s − 8.87·17-s + ⋯
L(s)  = 1  + (−0.0818 − 0.996i)2-s − 0.520i·3-s + (−0.986 + 0.163i)4-s + 0.687·5-s + (−0.518 + 0.0425i)6-s + 0.377i·7-s + (0.243 + 0.969i)8-s + 0.729·9-s + (−0.0562 − 0.685i)10-s + 0.771i·11-s + (0.0848 + 0.513i)12-s − 1.42·13-s + (0.376 − 0.0309i)14-s − 0.357i·15-s + (0.946 − 0.321i)16-s − 0.522·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.706467 - 0.599257i\)
\(L(\frac12)\) \(\approx\) \(0.706467 - 0.599257i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.163 + 1.99i)T \)
7 \( 1 - 2.64iT \)
good3 \( 1 + 1.56iT - 9T^{2} \)
5 \( 1 - 3.43T + 25T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 + 18.5T + 169T^{2} \)
17 \( 1 + 8.87T + 289T^{2} \)
19 \( 1 + 30.3iT - 361T^{2} \)
23 \( 1 - 26.5iT - 529T^{2} \)
29 \( 1 - 18.6T + 841T^{2} \)
31 \( 1 - 41.2iT - 961T^{2} \)
37 \( 1 + 3.49T + 1.36e3T^{2} \)
41 \( 1 - 37.7T + 1.68e3T^{2} \)
43 \( 1 + 50.8iT - 1.84e3T^{2} \)
47 \( 1 + 51.9iT - 2.20e3T^{2} \)
53 \( 1 - 15.3T + 2.80e3T^{2} \)
59 \( 1 + 38.3iT - 3.48e3T^{2} \)
61 \( 1 + 72.5T + 3.72e3T^{2} \)
67 \( 1 - 32.0iT - 4.48e3T^{2} \)
71 \( 1 - 50.6iT - 5.04e3T^{2} \)
73 \( 1 + 5.48T + 5.32e3T^{2} \)
79 \( 1 + 39.8iT - 6.24e3T^{2} \)
83 \( 1 + 4.28iT - 6.88e3T^{2} \)
89 \( 1 - 123.T + 7.92e3T^{2} \)
97 \( 1 + 32.0T + 9.40e3T^{2} \)
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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

−17.52103065367339427104940124679, −15.36693923048420988329348081390, −13.84105246619912641141782427265, −12.84755382431198112581871324287, −11.88053544807426675719344222716, −10.18967094449228738159124188043, −9.203404833991545109694140593872, −7.22273304333977905966444118788, −4.90187186608116542117443072336, −2.18943478194181923973646081641, 4.44786693874506147779299708564, 6.12223369836848965921124709915, 7.75328382635326175735943251381, 9.471623627578727220924373679148, 10.34585787492284660047317628124, 12.66503136129944813034505453755, 13.94839587850568902204141974601, 14.90263072485402376771715522757, 16.25048873668409978272873322269, 16.97613190566524916007627839952

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