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Properties

Label 2-28-4.3-c4-0-6
Degree $2$
Conductor $28$
Sign $0.999 - 0.0241i$
Analytic cond. $2.89435$
Root an. cond. $1.70128$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.79 − 2.86i)2-s + 13.4i·3-s + (−0.386 − 15.9i)4-s + 36.3·5-s + (38.5 + 37.6i)6-s + 18.5i·7-s + (−46.8 − 43.5i)8-s − 100.·9-s + (101. − 103. i)10-s − 79.8i·11-s + (215. − 5.20i)12-s − 298.·13-s + (53.0 + 51.7i)14-s + 489. i·15-s + (−255. + 12.3i)16-s + 197.·17-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)2-s + 1.49i·3-s + (−0.0241 − 0.999i)4-s + 1.45·5-s + (1.07 + 1.04i)6-s + 0.377i·7-s + (−0.732 − 0.681i)8-s − 1.24·9-s + (1.01 − 1.03i)10-s − 0.660i·11-s + (1.49 − 0.0361i)12-s − 1.76·13-s + (0.270 + 0.264i)14-s + 2.17i·15-s + (−0.998 + 0.0482i)16-s + 0.683·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0241i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.999 - 0.0241i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ 0.999 - 0.0241i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.08614 + 0.0251708i\)
\(L(\frac12)\) \(\approx\) \(2.08614 + 0.0251708i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.79 + 2.86i)T \)
7 \( 1 - 18.5iT \)
good3 \( 1 - 13.4iT - 81T^{2} \)
5 \( 1 - 36.3T + 625T^{2} \)
11 \( 1 + 79.8iT - 1.46e4T^{2} \)
13 \( 1 + 298.T + 2.85e4T^{2} \)
17 \( 1 - 197.T + 8.35e4T^{2} \)
19 \( 1 + 162. iT - 1.30e5T^{2} \)
23 \( 1 + 217. iT - 2.79e5T^{2} \)
29 \( 1 + 328.T + 7.07e5T^{2} \)
31 \( 1 + 374. iT - 9.23e5T^{2} \)
37 \( 1 - 1.44e3T + 1.87e6T^{2} \)
41 \( 1 - 771.T + 2.82e6T^{2} \)
43 \( 1 - 1.28e3iT - 3.41e6T^{2} \)
47 \( 1 - 4.01e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.33e3T + 7.89e6T^{2} \)
59 \( 1 + 1.35e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.32e3T + 1.38e7T^{2} \)
67 \( 1 - 4.30e3iT - 2.01e7T^{2} \)
71 \( 1 - 612. iT - 2.54e7T^{2} \)
73 \( 1 - 5.72e3T + 2.83e7T^{2} \)
79 \( 1 + 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 - 94.8iT - 4.74e7T^{2} \)
89 \( 1 + 2.59e3T + 6.27e7T^{2} \)
97 \( 1 - 1.17e3T + 8.85e7T^{2} \)
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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

−16.31178924509067669899190324124, −14.86928678163301292329077473594, −14.21726002144893582500448375518, −12.74601025477663001337798727855, −11.12578685266990106819513435474, −9.877350001031736924234128281845, −9.430691440540328924908041147617, −5.84761725288246887424914307904, −4.77968665585901696764013078160, −2.71901359630462026309464560140, 2.19988335627176615951878632606, 5.38110708458159296478812475358, 6.73997469781511308734829908918, 7.70276868832353474278054245098, 9.723637304880260459893911739078, 12.13298150763629251447443613923, 12.92679675737450870721367677357, 13.90015969454952049799147009856, 14.70563214059677621704296560967, 16.89287879034458841843064549819

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