Properties

Label 2-300-3.2-c4-0-7
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 71·7-s + 81·9-s − 191·13-s − 601·19-s + 639·21-s − 729·27-s + 1.55e3·31-s + 2.06e3·37-s + 1.71e3·39-s − 3.19e3·43-s + 2.64e3·49-s + 5.40e3·57-s + 7.19e3·61-s − 5.75e3·63-s + 8.80e3·67-s + 8.54e3·73-s + 7.68e3·79-s + 6.56e3·81-s + 1.35e4·91-s − 1.40e4·93-s − 9.07e3·97-s − 1.64e4·103-s − 1.87e4·109-s − 1.85e4·111-s − 1.54e4·117-s + ⋯
L(s)  = 1  − 3-s − 1.44·7-s + 9-s − 1.13·13-s − 1.66·19-s + 1.44·21-s − 27-s + 1.62·31-s + 1.50·37-s + 1.13·39-s − 1.72·43-s + 1.09·49-s + 1.66·57-s + 1.93·61-s − 1.44·63-s + 1.96·67-s + 1.60·73-s + 1.23·79-s + 81-s + 1.63·91-s − 1.62·93-s − 0.964·97-s − 1.54·103-s − 1.57·109-s − 1.50·111-s − 1.13·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6726011457\)
\(L(\frac12)\) \(\approx\) \(0.6726011457\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p^{2} T \)
5 \( 1 \)
good7 \( 1 + 71 T + p^{4} T^{2} \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 + 191 T + p^{4} T^{2} \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( 1 + 601 T + p^{4} T^{2} \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 - 1559 T + p^{4} T^{2} \)
37 \( 1 - 2062 T + p^{4} T^{2} \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 + 3191 T + p^{4} T^{2} \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 - 7199 T + p^{4} T^{2} \)
67 \( 1 - 8809 T + p^{4} T^{2} \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( 1 - 8542 T + p^{4} T^{2} \)
79 \( 1 - 7682 T + p^{4} T^{2} \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 + 9071 T + p^{4} T^{2} \)
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

−11.09983734630539392727539559543, −10.03624016075824611514470287854, −9.653110739828543612089462821927, −8.164087184168340754654666060898, −6.75670464424963822729722870302, −6.39779787011724144417460469437, −5.10568194631890160253944543146, −3.98968107275729611086561496948, −2.45615454895435937753756402893, −0.50786256077308963817178164574, 0.50786256077308963817178164574, 2.45615454895435937753756402893, 3.98968107275729611086561496948, 5.10568194631890160253944543146, 6.39779787011724144417460469437, 6.75670464424963822729722870302, 8.164087184168340754654666060898, 9.653110739828543612089462821927, 10.03624016075824611514470287854, 11.09983734630539392727539559543

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