Properties

Label 2-70-1.1-c5-0-0
Degree $2$
Conductor $70$
Sign $1$
Analytic cond. $11.2268$
Root an. cond. $3.35065$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 49·7-s − 64·8-s − 162·9-s − 100·10-s − 187·11-s − 144·12-s + 627·13-s + 196·14-s − 225·15-s + 256·16-s + 1.81e3·17-s + 648·18-s + 258·19-s + 400·20-s + 441·21-s + 748·22-s + 2.97e3·23-s + 576·24-s + 625·25-s − 2.50e3·26-s + 3.64e3·27-s − 784·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.465·11-s − 0.288·12-s + 1.02·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.52·17-s + 0.471·18-s + 0.163·19-s + 0.223·20-s + 0.218·21-s + 0.329·22-s + 1.17·23-s + 0.204·24-s + 1/5·25-s − 0.727·26-s + 0.962·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(11.2268\)
Root analytic conductor: \(3.35065\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.017051004\)
\(L(\frac12)\) \(\approx\) \(1.017051004\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
5 \( 1 - p^{2} T \)
7 \( 1 + p^{2} T \)
good3 \( 1 + p^{2} T + p^{5} T^{2} \)
11 \( 1 + 17 p T + p^{5} T^{2} \)
13 \( 1 - 627 T + p^{5} T^{2} \)
17 \( 1 - 1813 T + p^{5} T^{2} \)
19 \( 1 - 258 T + p^{5} T^{2} \)
23 \( 1 - 2970 T + p^{5} T^{2} \)
29 \( 1 - 1299 T + p^{5} T^{2} \)
31 \( 1 - 1916 T + p^{5} T^{2} \)
37 \( 1 - 6578 T + p^{5} T^{2} \)
41 \( 1 - 6676 T + p^{5} T^{2} \)
43 \( 1 - 3178 T + p^{5} T^{2} \)
47 \( 1 + 22001 T + p^{5} T^{2} \)
53 \( 1 - 26168 T + p^{5} T^{2} \)
59 \( 1 - 3932 T + p^{5} T^{2} \)
61 \( 1 + 48740 T + p^{5} T^{2} \)
67 \( 1 + 44832 T + p^{5} T^{2} \)
71 \( 1 - 63736 T + p^{5} T^{2} \)
73 \( 1 - 60470 T + p^{5} T^{2} \)
79 \( 1 + 43721 T + p^{5} T^{2} \)
83 \( 1 - 1172 p T + p^{5} T^{2} \)
89 \( 1 - 45560 T + p^{5} T^{2} \)
97 \( 1 + 57295 T + p^{5} T^{2} \)
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

−13.66622230802704197590456070706, −12.43080389466818216092553098836, −11.26871829003105819032180345463, −10.36202315317844147617136591421, −9.196018945458646807224067532598, −7.966202135995660929490627386416, −6.42376460321667965437252922449, −5.42684175191597055050200631682, −3.03091307706779167125837789824, −0.924719863083208071621048271525, 0.924719863083208071621048271525, 3.03091307706779167125837789824, 5.42684175191597055050200631682, 6.42376460321667965437252922449, 7.966202135995660929490627386416, 9.196018945458646807224067532598, 10.36202315317844147617136591421, 11.26871829003105819032180345463, 12.43080389466818216092553098836, 13.66622230802704197590456070706

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