↑

Properties

Label 2-28-7.6-c2-0-0
Degree $2$
Conductor $28$
Sign $0.714 - 0.699i$
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  + 4.89i·3-s − 4.89i·5-s + (5 − 4.89i)7-s − 14.9·9-s − 6·11-s + 4.89i·13-s + 23.9·15-s − 19.5i·17-s + 24.4i·19-s + (23.9 + 24.4i)21-s − 30·23-s + 1.00·25-s − 29.3i·27-s − 6·29-s − 29.3i·33-s + ⋯
L(s)  = 1  + 1.63i·3-s − 0.979i·5-s + (0.714 − 0.699i)7-s − 1.66·9-s − 0.545·11-s + 0.376i·13-s + 1.59·15-s − 1.15i·17-s + 1.28i·19-s + (1.14 + 1.16i)21-s − 1.30·23-s + 0.0400·25-s − 1.08i·27-s − 0.206·29-s − 0.890i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.714 - 0.699i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.896934 + 0.366172i\)
\(L(\frac12)\) \(\approx\) \(0.896934 + 0.366172i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-5 + 4.89i)T \)
good3 \( 1 - 4.89iT - 9T^{2} \)
5 \( 1 + 4.89iT - 25T^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 - 4.89iT - 169T^{2} \)
17 \( 1 + 19.5iT - 289T^{2} \)
19 \( 1 - 24.4iT - 361T^{2} \)
23 \( 1 + 30T + 529T^{2} \)
29 \( 1 + 6T + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 10T + 1.36e3T^{2} \)
41 \( 1 - 48.9iT - 1.68e3T^{2} \)
43 \( 1 - 10T + 1.84e3T^{2} \)
47 \( 1 + 19.5iT - 2.20e3T^{2} \)
53 \( 1 - 90T + 2.80e3T^{2} \)
59 \( 1 + 24.4iT - 3.48e3T^{2} \)
61 \( 1 - 24.4iT - 3.72e3T^{2} \)
67 \( 1 + 70T + 4.48e3T^{2} \)
71 \( 1 - 42T + 5.04e3T^{2} \)
73 \( 1 + 107. iT - 5.32e3T^{2} \)
79 \( 1 - 74T + 6.24e3T^{2} \)
83 \( 1 - 63.6iT - 6.88e3T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 - 78.3iT - 9.40e3T^{2} \)
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

−16.55687662739985045994637523519, −16.32510798008001579172429449035, −14.88408523470219201138873081677, −13.74088378107400828707586943021, −11.88210003694274100340271822057, −10.54152099235473083826173450415, −9.482227615847877260948823674436, −8.102851334220104601956306921892, −5.23974565656898765763989074898, −4.14483405066882728762060739027, 2.35788931264915256215796178855, 5.92187780477507538380273482265, 7.28194112582239693475608969263, 8.414074673860621653697820224068, 10.76355883241477193299422174829, 11.97016493292561868322426830582, 13.10212930056448555387090013297, 14.28217893432124476374902789828, 15.36279602132048395755420746153, 17.49546827430041657887027019916

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