Properties

Label 2-28-4.3-c2-0-2
Degree $2$
Conductor $28$
Sign $0.525 + 0.851i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.92 + 0.545i)2-s − 4.54i·3-s + (3.40 − 2.10i)4-s + 1.36·5-s + (2.48 + 8.74i)6-s − 2.64i·7-s + (−5.40 + 5.89i)8-s − 11.6·9-s + (−2.63 + 0.746i)10-s + 15.3i·11-s + (−9.54 − 15.4i)12-s + 19.9·13-s + (1.44 + 5.09i)14-s − 6.21i·15-s + (7.17 − 14.2i)16-s − 4.73·17-s + ⋯
L(s)  = 1  + (−0.962 + 0.272i)2-s − 1.51i·3-s + (0.851 − 0.525i)4-s + 0.273·5-s + (0.413 + 1.45i)6-s − 0.377i·7-s + (−0.675 + 0.737i)8-s − 1.29·9-s + (−0.263 + 0.0746i)10-s + 1.39i·11-s + (−0.795 − 1.28i)12-s + 1.53·13-s + (0.103 + 0.363i)14-s − 0.414i·15-s + (0.448 − 0.893i)16-s − 0.278·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.525 + 0.851i$
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: \((2,\ 28,\ (\ :1),\ 0.525 + 0.851i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.602071 - 0.336011i\)
\(L(\frac12)\) \(\approx\) \(0.602071 - 0.336011i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 - 0.545i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 4.54iT - 9T^{2} \)
5 \( 1 - 1.36T + 25T^{2} \)
11 \( 1 - 15.3iT - 121T^{2} \)
13 \( 1 - 19.9T + 169T^{2} \)
17 \( 1 + 4.73T + 289T^{2} \)
19 \( 1 - 13.2iT - 361T^{2} \)
23 \( 1 + 14.9iT - 529T^{2} \)
29 \( 1 - 6.20T + 841T^{2} \)
31 \( 1 - 18.5iT - 961T^{2} \)
37 \( 1 + 27.5T + 1.36e3T^{2} \)
41 \( 1 + 11.1T + 1.68e3T^{2} \)
43 \( 1 - 27.0iT - 1.84e3T^{2} \)
47 \( 1 + 30.9iT - 2.20e3T^{2} \)
53 \( 1 + 12.7T + 2.80e3T^{2} \)
59 \( 1 + 28.8iT - 3.48e3T^{2} \)
61 \( 1 - 6.52T + 3.72e3T^{2} \)
67 \( 1 - 102. iT - 4.48e3T^{2} \)
71 \( 1 - 45.8iT - 5.04e3T^{2} \)
73 \( 1 + 70.0T + 5.32e3T^{2} \)
79 \( 1 + 33.3iT - 6.24e3T^{2} \)
83 \( 1 + 159. iT - 6.88e3T^{2} \)
89 \( 1 + 50.0T + 7.92e3T^{2} \)
97 \( 1 - 89.7T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.34439056618848207821870331104, −15.86274734450940922703739786063, −14.31253467648419788916335462958, −13.03288612571902477814342810913, −11.77317084631431915866697327411, −10.23174790777163133091244438011, −8.507554387749978695081078152812, −7.28675889816114551751227062113, −6.22775598100866860173054403083, −1.68304587108463664026036144013, 3.50379472561579913845785116931, 5.94205506762712360266347020754, 8.494284247850643041540154761718, 9.359917243644393456362464360202, 10.70228510988451336569799943121, 11.43870571015080607516480459093, 13.60061345888721301558260892673, 15.47518231717247330750589008081, 15.97954147200388943795178546990, 17.07050026521147063909929678090

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