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 − 3.23·7-s + 8-s + 9-s − 10-s − 12-s + 3.38·13-s − 3.23·14-s + 15-s + 16-s − 6.09·17-s + 18-s + 1.23·19-s − 20-s + 3.23·21-s + 6.85·23-s − 24-s + 25-s + 3.38·26-s − 27-s − 3.23·28-s − 2.61·29-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 − 1.22·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 0.937·13-s − 0.864·14-s + 0.258·15-s + 0.250·16-s − 1.47·17-s + 0.235·18-s + 0.283·19-s − 0.223·20-s + 0.706·21-s + 1.42·23-s − 0.204·24-s + 0.200·25-s + 0.663·26-s − 0.192·27-s − 0.611·28-s − 0.486·29-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

\(L(1)\) \(\approx\) \(1.724724328\)
\(L(\frac12)\) \(\approx\) \(1.724724328\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23T + 7T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 + 6.09T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
29 \( 1 + 2.61T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 + 6.38T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 + 0.0901T + 43T^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 4.09T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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

−8.663664574857548568501955385403, −7.50141446340011959048795912227, −6.72483259520958704336880896900, −6.46283570619403572495136915072, −5.50173183784178937384952316295, −4.78583037474975579078024395617, −3.80183146455483967126646690612, −3.32614083794030071709771394504, −2.16509272387993420225446256538, −0.69559860921755579442895495884, 0.69559860921755579442895495884, 2.16509272387993420225446256538, 3.32614083794030071709771394504, 3.80183146455483967126646690612, 4.78583037474975579078024395617, 5.50173183784178937384952316295, 6.46283570619403572495136915072, 6.72483259520958704336880896900, 7.50141446340011959048795912227, 8.663664574857548568501955385403

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