Properties

Degree $2$
Conductor $3630$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 4.49·7-s + 8-s + 9-s − 10-s + 12-s + 2.33·13-s + 4.49·14-s − 15-s + 16-s + 5.15·17-s + 18-s + 6.01·19-s − 20-s + 4.49·21-s − 8.88·23-s + 24-s + 25-s + 2.33·26-s + 27-s + 4.49·28-s − 7.58·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.69·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 0.647·13-s + 1.20·14-s − 0.258·15-s + 0.250·16-s + 1.25·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.980·21-s − 1.85·23-s + 0.204·24-s + 0.200·25-s + 0.457·26-s + 0.192·27-s + 0.848·28-s − 1.40·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3630} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.607627181\)
\(L(\frac12)\) \(\approx\) \(4.607627181\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 - 4.49T + 7T^{2} \)
13 \( 1 - 2.33T + 13T^{2} \)
17 \( 1 - 5.15T + 17T^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 - 7.93T + 37T^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 - 0.0785T + 43T^{2} \)
47 \( 1 - 2.90T + 47T^{2} \)
53 \( 1 - 0.745T + 53T^{2} \)
59 \( 1 + 5.32T + 59T^{2} \)
61 \( 1 + 15.4T + 61T^{2} \)
67 \( 1 + 1.67T + 67T^{2} \)
71 \( 1 - 9.55T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 5.47T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + 7.41T + 89T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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

−8.233260014622122496622909266098, −7.70709289251031762452506774997, −7.46083362121663760684224327327, −6.08454709983431390268044812026, −5.43634951923656523618602256567, −4.66448898325768985640712959115, −3.87216660369152442103566152108, −3.24140481015611356322499984565, −1.99955995047539241488495197383, −1.26729446496382611540885515928, 1.26729446496382611540885515928, 1.99955995047539241488495197383, 3.24140481015611356322499984565, 3.87216660369152442103566152108, 4.66448898325768985640712959115, 5.43634951923656523618602256567, 6.08454709983431390268044812026, 7.46083362121663760684224327327, 7.70709289251031762452506774997, 8.233260014622122496622909266098

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