↑

Properties

Label 2-1500-25.9-c1-0-7
Degree $2$
Conductor $1500$
Sign $0.995 + 0.0968i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)3-s − 0.595i·7-s + (0.809 − 0.587i)9-s + (−2.71 − 1.97i)11-s + (2.80 + 3.85i)13-s + (7.11 + 2.31i)17-s + (−1.91 + 5.88i)19-s + (−0.184 − 0.566i)21-s + (2.59 − 3.56i)23-s + (0.587 − 0.809i)27-s + (0.853 + 2.62i)29-s + (1.38 − 4.26i)31-s + (−3.19 − 1.03i)33-s + (0.764 + 1.05i)37-s + (3.85 + 2.80i)39-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s − 0.225i·7-s + (0.269 − 0.195i)9-s + (−0.818 − 0.594i)11-s + (0.777 + 1.07i)13-s + (1.72 + 0.560i)17-s + (−0.438 + 1.35i)19-s + (−0.0401 − 0.123i)21-s + (0.540 − 0.743i)23-s + (0.113 − 0.155i)27-s + (0.158 + 0.487i)29-s + (0.249 − 0.766i)31-s + (−0.555 − 0.180i)33-s + (0.125 + 0.172i)37-s + (0.617 + 0.448i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.995 + 0.0968i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.995 + 0.0968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157531002\)
\(L(\frac12)\) \(\approx\) \(2.157531002\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good7 \( 1 + 0.595iT - 7T^{2} \)
11 \( 1 + (2.71 + 1.97i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.80 - 3.85i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-7.11 - 2.31i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.91 - 5.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.59 + 3.56i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.853 - 2.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.38 + 4.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.764 - 1.05i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.61 + 5.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.59iT - 43T^{2} \)
47 \( 1 + (-4.18 + 1.36i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.80 + 2.53i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.80 - 1.31i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (10.2 + 7.41i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (7.94 + 2.58i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.09 - 6.46i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.29 - 5.91i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.26 - 10.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.97 - 1.29i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-3.43 - 2.49i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-11.6 + 3.79i)T + (78.4 - 57.0i)T^{2} \)
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

−9.358987694014869890751182854072, −8.556822902222338145293506044647, −7.980925317371178294270628649248, −7.20061507368082765377249169095, −6.14865558496731155262515733081, −5.50283260988360191787844578555, −4.13202038673246022626936533272, −3.50892658308648173367922662587, −2.32970992677919756242947360786, −1.11060593320669434137350131612, 1.05980301119483044664347840396, 2.66904159810249108675190621791, 3.18739211467013350675201740295, 4.49155010324003195032810149104, 5.30667295139790793527720418113, 6.12074215270119735153284903725, 7.50442737734237565860215839697, 7.69694855095639517262005530476, 8.776121649485216219120780313199, 9.416107170763942886373189934804

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