Properties

Label 2-5070-13.12-c1-0-25
Degree $2$
Conductor $5070$
Sign $-0.691 - 0.722i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 3-s − 4-s i·5-s + i·6-s + 0.643i·7-s i·8-s + 9-s + 10-s + 4.40i·11-s − 12-s − 0.643·14-s i·15-s + 16-s − 0.939·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.243i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.32i·11-s − 0.288·12-s − 0.171·14-s − 0.258i·15-s + 0.250·16-s − 0.227·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.691 - 0.722i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.759578350\)
\(L(\frac12)\) \(\approx\) \(1.759578350\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.643iT - 7T^{2} \)
11 \( 1 - 4.40iT - 11T^{2} \)
17 \( 1 + 0.939T + 17T^{2} \)
19 \( 1 + 2.75iT - 19T^{2} \)
23 \( 1 - 2.04T + 23T^{2} \)
29 \( 1 + 0.149T + 29T^{2} \)
31 \( 1 + 2.08iT - 31T^{2} \)
37 \( 1 - 6.07iT - 37T^{2} \)
41 \( 1 - 5.74iT - 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 7.85iT - 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 1.45iT - 67T^{2} \)
71 \( 1 - 7.03iT - 71T^{2} \)
73 \( 1 - 14.8iT - 73T^{2} \)
79 \( 1 - 0.929T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 7.47iT - 89T^{2} \)
97 \( 1 - 5.05iT - 97T^{2} \)
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

−8.527732083008923415566087467315, −7.69255959017892427861800085908, −7.18815131371889765224929917862, −6.47257167863705888701548148717, −5.60514136865861696400643970732, −4.62893893186490543550712198587, −4.43785907864608096397679481531, −3.20418376365781325115226018298, −2.28313307707080526488448093929, −1.21260996413552044409472272698, 0.45164107774321531685043783503, 1.66638891245132606962745416442, 2.58572187856610350622794271558, 3.44973874229984704262759082524, 3.80762054526566905777518459465, 4.91404753015908897957852061160, 5.75140228574212187561810992811, 6.53430779062340565228653116284, 7.41105481113175808164358406259, 8.077143688341806538437001525105

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