Properties

Label 2-675-1.1-c3-0-49
Degree $2$
Conductor $675$
Sign $-1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 17·7-s + 70·13-s + 64·16-s + 107·19-s + 136·28-s − 289·31-s − 323·37-s − 71·43-s − 54·49-s − 560·52-s − 901·61-s − 512·64-s + 880·67-s + 919·73-s − 856·76-s − 1.38e3·79-s − 1.19e3·91-s − 1.85e3·97-s + 1.80e3·103-s − 1.56e3·109-s − 1.08e3·112-s + ⋯
L(s)  = 1  − 4-s − 0.917·7-s + 1.49·13-s + 16-s + 1.29·19-s + 0.917·28-s − 1.67·31-s − 1.43·37-s − 0.251·43-s − 0.157·49-s − 1.49·52-s − 1.89·61-s − 64-s + 1.60·67-s + 1.47·73-s − 1.29·76-s − 1.97·79-s − 1.37·91-s − 1.93·97-s + 1.72·103-s − 1.37·109-s − 0.917·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + p^{3} T^{2} \)
7 \( 1 + 17 T + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 - 70 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 - 107 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 + 289 T + p^{3} T^{2} \)
37 \( 1 + 323 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 71 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 901 T + p^{3} T^{2} \)
67 \( 1 - 880 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 - 919 T + p^{3} T^{2} \)
79 \( 1 + 1387 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 1853 T + p^{3} T^{2} \)
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

−9.511014017572136753001734678425, −8.967529609515399255111202323022, −8.081324869396578531683138568182, −6.99803821873125080443977344778, −5.93712079797926490478855752544, −5.16956989363084313614834467004, −3.82426391502226617479874054683, −3.27247442286714994876927027987, −1.33231959963257179621788545111, 0, 1.33231959963257179621788545111, 3.27247442286714994876927027987, 3.82426391502226617479874054683, 5.16956989363084313614834467004, 5.93712079797926490478855752544, 6.99803821873125080443977344778, 8.081324869396578531683138568182, 8.967529609515399255111202323022, 9.511014017572136753001734678425

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