Properties

Label 2-5070-13.12-c1-0-64
Degree $2$
Conductor $5070$
Sign $0.832 + 0.554i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
Motivic weight $1$
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  + i·2-s − 3-s − 4-s + i·5-s i·6-s i·8-s + 9-s − 10-s − 4i·11-s + 12-s i·15-s + 16-s + 6·17-s + i·18-s + 4i·19-s i·20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.20i·11-s + 0.288·12-s − 0.258i·15-s + 0.250·16-s + 1.45·17-s + 0.235i·18-s + 0.917i·19-s − 0.223i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9824146383\)
\(L(\frac12)\) \(\approx\) \(0.9824146383\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
5 \( 1 - iT \)
13 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 16iT - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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

−8.072109998916235197105837615063, −7.50233812875175255108011869558, −6.53871494493932803496032202455, −5.90255786825983424374140417080, −5.67281168118077168719861733110, −4.56725924546548230116991366418, −3.73093990139767201189774484145, −3.01022719892157244847149251919, −1.56538039233603013895554301316, −0.34515317747471623257156338591, 0.985622654308429601258639247182, 1.85225113094165631182200785856, 2.86520048737423861042903124174, 3.90967077195863446539368182483, 4.60823928175496984421791453168, 5.21464337660547116155464114702, 5.97208297641197493495612767500, 6.89345169608282922067128485984, 7.63143783172067378376054679059, 8.322744553475654579019178608893

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