Properties

Label 2-3675-1.1-c1-0-96
Degree $2$
Conductor $3675$
Sign $-1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.185·2-s + 3-s − 1.96·4-s − 0.185·6-s + 0.737·8-s + 9-s − 5.19·11-s − 1.96·12-s − 2.77·13-s + 3.79·16-s + 2.56·17-s − 0.185·18-s + 5.08·19-s + 0.965·22-s + 2.09·23-s + 0.737·24-s + 0.516·26-s + 27-s + 4.66·29-s + 3.49·31-s − 2.17·32-s − 5.19·33-s − 0.476·34-s − 1.96·36-s − 6.20·37-s − 0.944·38-s − 2.77·39-s + ⋯
L(s)  = 1  − 0.131·2-s + 0.577·3-s − 0.982·4-s − 0.0758·6-s + 0.260·8-s + 0.333·9-s − 1.56·11-s − 0.567·12-s − 0.770·13-s + 0.948·16-s + 0.622·17-s − 0.0438·18-s + 1.16·19-s + 0.205·22-s + 0.436·23-s + 0.150·24-s + 0.101·26-s + 0.192·27-s + 0.866·29-s + 0.628·31-s − 0.385·32-s − 0.904·33-s − 0.0817·34-s − 0.327·36-s − 1.01·37-s − 0.153·38-s − 0.445·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(29.3450\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + 0.185T + 2T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 2.77T + 13T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 - 5.08T + 19T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 - 4.66T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 + 6.20T + 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 4.15T + 53T^{2} \)
59 \( 1 - 0.997T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 3.97T + 73T^{2} \)
79 \( 1 - 3.23T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 4.68T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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

−8.206322783666438472231420407662, −7.61384368096933709989554097319, −6.98251696532181665380114185873, −5.58713382877263590137517566069, −5.12397465568752355204684759744, −4.42528238133269536958100439067, −3.26585353354353028210891026916, −2.75521154273470316594977887855, −1.35342311551794166379461870858, 0, 1.35342311551794166379461870858, 2.75521154273470316594977887855, 3.26585353354353028210891026916, 4.42528238133269536958100439067, 5.12397465568752355204684759744, 5.58713382877263590137517566069, 6.98251696532181665380114185873, 7.61384368096933709989554097319, 8.206322783666438472231420407662

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