Properties

Label 2-105-5.4-c3-0-2
Degree $2$
Conductor $105$
Sign $0.861 - 0.508i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4.88i·2-s + 3i·3-s − 15.9·4-s + (−9.63 + 5.67i)5-s + 14.6·6-s + 7i·7-s + 38.6i·8-s − 9·9-s + (27.7 + 47.0i)10-s + 54.9·11-s − 47.7i·12-s + 49.7i·13-s + 34.2·14-s + (−17.0 − 28.8i)15-s + 61.7·16-s + 133. i·17-s + ⋯
L(s)  = 1  − 1.72i·2-s + 0.577i·3-s − 1.98·4-s + (−0.861 + 0.508i)5-s + 0.998·6-s + 0.377i·7-s + 1.70i·8-s − 0.333·9-s + (0.878 + 1.48i)10-s + 1.50·11-s − 1.14i·12-s + 1.06i·13-s + 0.653·14-s + (−0.293 − 0.497i)15-s + 0.964·16-s + 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.861 - 0.508i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.755842 + 0.206288i\)
\(L(\frac12)\) \(\approx\) \(0.755842 + 0.206288i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 + (9.63 - 5.67i)T \)
7 \( 1 - 7iT \)
good2 \( 1 + 4.88iT - 8T^{2} \)
11 \( 1 - 54.9T + 1.33e3T^{2} \)
13 \( 1 - 49.7iT - 2.19e3T^{2} \)
17 \( 1 - 133. iT - 4.91e3T^{2} \)
19 \( 1 + 138.T + 6.85e3T^{2} \)
23 \( 1 + 7.32iT - 1.21e4T^{2} \)
29 \( 1 + 87.2T + 2.43e4T^{2} \)
31 \( 1 + 209.T + 2.97e4T^{2} \)
37 \( 1 + 67.9iT - 5.06e4T^{2} \)
41 \( 1 - 77.6T + 6.89e4T^{2} \)
43 \( 1 - 197. iT - 7.95e4T^{2} \)
47 \( 1 - 4.97iT - 1.03e5T^{2} \)
53 \( 1 - 53.0iT - 1.48e5T^{2} \)
59 \( 1 - 683.T + 2.05e5T^{2} \)
61 \( 1 + 26.8T + 2.26e5T^{2} \)
67 \( 1 - 149. iT - 3.00e5T^{2} \)
71 \( 1 - 6.15T + 3.57e5T^{2} \)
73 \( 1 + 294. iT - 3.89e5T^{2} \)
79 \( 1 - 938.T + 4.93e5T^{2} \)
83 \( 1 + 784. iT - 5.71e5T^{2} \)
89 \( 1 + 275.T + 7.04e5T^{2} \)
97 \( 1 - 1.16e3iT - 9.12e5T^{2} \)
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

−12.88700374079946699487414139962, −12.03772515401801868329612376679, −11.20932378825912725843916749142, −10.57109110606400198242669912792, −9.275927268385660273364778970614, −8.524443798674431410728837745220, −6.44236979367788911178769130853, −4.19181882194962807505477648783, −3.77129916583061970201418224630, −1.93752015487799232672000412063, 0.44928666203797719394675372730, 3.99974671043869865632450468572, 5.30095993833632765420276781059, 6.69644002191792149335098483460, 7.42330824102590640755743711813, 8.442808553472678563399484452737, 9.304928023744878854175232108392, 11.27743264105076338988419045449, 12.48698607219363729038366796433, 13.45599382062213503484712799753

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