↑

Properties

Label 2-3675-735.254-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.798 - 0.601i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.149 + 0.988i)3-s + (−0.0747 − 0.997i)4-s + (0.294 + 0.955i)7-s + (−0.955 − 0.294i)9-s + (0.997 + 0.0747i)12-s + (0.712 + 0.162i)13-s + (−0.988 + 0.149i)16-s + (0.955 − 1.65i)19-s + (−0.988 + 0.149i)21-s + (0.433 − 0.900i)27-s + (0.930 − 0.365i)28-s + (0.900 + 1.56i)31-s + (−0.222 + 0.974i)36-s + (1.64 + 0.123i)37-s + (−0.266 + 0.680i)39-s + ⋯
L(s)  = 1  + (−0.149 + 0.988i)3-s + (−0.0747 − 0.997i)4-s + (0.294 + 0.955i)7-s + (−0.955 − 0.294i)9-s + (0.997 + 0.0747i)12-s + (0.712 + 0.162i)13-s + (−0.988 + 0.149i)16-s + (0.955 − 1.65i)19-s + (−0.988 + 0.149i)21-s + (0.433 − 0.900i)27-s + (0.930 − 0.365i)28-s + (0.900 + 1.56i)31-s + (−0.222 + 0.974i)36-s + (1.64 + 0.123i)37-s + (−0.266 + 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.798 - 0.601i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.798 - 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243883659\)
\(L(\frac12)\) \(\approx\) \(1.243883659\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.149 - 0.988i)T \)
5 \( 1 \)
7 \( 1 + (-0.294 - 0.955i)T \)
good2 \( 1 + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.712 - 0.162i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.955 + 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.64 - 0.123i)T + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.347 - 0.277i)T + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (-0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.0111 + 0.149i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (1.71 - 0.988i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.12 - 1.21i)T + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 1.46iT - T^{2} \)
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

−8.802833737254279459319680596156, −8.517876770624465678709474493588, −7.18163462821784810568912959562, −6.23941525644120560920778568974, −5.76441423057566934366985891262, −4.86184483868637076269224776148, −4.61059081453536236933806285572, −3.24631820292007613581335675558, −2.47399855965122704059981026372, −1.09212850668704230254950859038, 0.942150851897989654065633968903, 2.06094172582086657611096684178, 3.18455836855925191726124385955, 3.85733610442842810901821149615, 4.75667826039069306679469709639, 5.91515010800582241089358437653, 6.43815159824325681953705248085, 7.49999383369978660502135637782, 7.76715417533230305260643977365, 8.244491548947099573064068094357

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