↑

Properties

Label 2-300-5.4-c3-0-0
Degree $2$
Conductor $300$
Sign $-0.447 - 0.894i$
Analytic cond. $17.7005$
Root an. cond. $4.20720$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 7i·7-s − 9·9-s − 54·11-s + 55i·13-s + 18i·17-s + 25·19-s − 21·21-s + 18i·23-s + 27i·27-s + 54·29-s − 271·31-s + 162i·33-s + 314i·37-s + 165·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 1.48·11-s + 1.17i·13-s + 0.256i·17-s + 0.301·19-s − 0.218·21-s + 0.163i·23-s + 0.192i·27-s + 0.345·29-s − 1.57·31-s + 0.854i·33-s + 1.39i·37-s + 0.677·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(17.7005\)
Root analytic conductor: \(4.20720\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4983210546\)
\(L(\frac12)\) \(\approx\) \(0.4983210546\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 + 7iT - 343T^{2} \)
11 \( 1 + 54T + 1.33e3T^{2} \)
13 \( 1 - 55iT - 2.19e3T^{2} \)
17 \( 1 - 18iT - 4.91e3T^{2} \)
19 \( 1 - 25T + 6.85e3T^{2} \)
23 \( 1 - 18iT - 1.21e4T^{2} \)
29 \( 1 - 54T + 2.43e4T^{2} \)
31 \( 1 + 271T + 2.97e4T^{2} \)
37 \( 1 - 314iT - 5.06e4T^{2} \)
41 \( 1 + 360T + 6.89e4T^{2} \)
43 \( 1 - 163iT - 7.95e4T^{2} \)
47 \( 1 - 522iT - 1.03e5T^{2} \)
53 \( 1 - 36iT - 1.48e5T^{2} \)
59 \( 1 + 126T + 2.05e5T^{2} \)
61 \( 1 - 47T + 2.26e5T^{2} \)
67 \( 1 + 343iT - 3.00e5T^{2} \)
71 \( 1 + 1.08e3T + 3.57e5T^{2} \)
73 \( 1 - 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 - 568T + 4.93e5T^{2} \)
83 \( 1 + 1.42e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.44e3T + 7.04e5T^{2} \)
97 \( 1 + 439iT - 9.12e5T^{2} \)
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

−11.60609884591652711900355878266, −10.75672836086343634244033158448, −9.783198211330513501265108725625, −8.634771083437994388019404849359, −7.67795466423533824726446588063, −6.87690304715967244055799843585, −5.69424975341897271626081421592, −4.53140699414763792052279030522, −3.00958633973042921810540071001, −1.62210006092670562822078350465, 0.17602046240831288272219351373, 2.42987144276404713890696522463, 3.55770088522406099045002116059, 5.14037377627294060158795383127, 5.62510119856003649197402528406, 7.25619094007850076495064708602, 8.189119290841646359411591304701, 9.126702531666962289188497627986, 10.29633388406854465694486111123, 10.70637666713420104915664190723

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