↑

Properties

Label 2-1500-25.19-c1-0-5
Degree $2$
Conductor $1500$
Sign $0.0256 - 0.999i$
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  + (0.587 + 0.809i)3-s + 0.0883i·7-s + (−0.309 + 0.951i)9-s + (0.701 + 2.15i)11-s + (2.52 + 0.819i)13-s + (−1.22 + 1.68i)17-s + (1.42 + 1.03i)19-s + (−0.0714 + 0.0519i)21-s + (−4.50 + 1.46i)23-s + (−0.951 + 0.309i)27-s + (2.99 − 2.17i)29-s + (3.32 + 2.41i)31-s + (−1.33 + 1.83i)33-s + (−6.77 − 2.19i)37-s + (0.819 + 2.52i)39-s + ⋯
L(s)  = 1  + (0.339 + 0.467i)3-s + 0.0333i·7-s + (−0.103 + 0.317i)9-s + (0.211 + 0.650i)11-s + (0.699 + 0.227i)13-s + (−0.297 + 0.409i)17-s + (0.326 + 0.237i)19-s + (−0.0155 + 0.0113i)21-s + (−0.940 + 0.305i)23-s + (−0.183 + 0.0594i)27-s + (0.556 − 0.404i)29-s + (0.596 + 0.433i)31-s + (−0.232 + 0.319i)33-s + (−1.11 − 0.361i)37-s + (0.131 + 0.403i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0256 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0256 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.0256 - 0.999i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.0256 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763465529\)
\(L(\frac12)\) \(\approx\) \(1.763465529\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 0.0883iT - 7T^{2} \)
11 \( 1 + (-0.701 - 2.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.52 - 0.819i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.22 - 1.68i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.42 - 1.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.50 - 1.46i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.99 + 2.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.32 - 2.41i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.77 + 2.19i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.03 - 6.26i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.79iT - 43T^{2} \)
47 \( 1 + (-5.94 - 8.17i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.565 - 0.777i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.77 + 8.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.92 - 9.00i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (8.17 - 11.2i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.97 - 3.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.00 + 0.975i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.8 + 7.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.29 - 12.7i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.16 - 9.74i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.52 + 10.3i)T + (-29.9 + 92.2i)T^{2} \)
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

−9.743823506596208470561136419832, −8.847741215128998555750922913941, −8.240541151173873679792315100895, −7.32871292161465092379349889704, −6.40381795696131315917034184145, −5.55659815491109533362412628973, −4.45343434919539631249296522926, −3.83170779055495357973650009291, −2.68165989442032169516340373272, −1.49011658685552120102544601022, 0.69817080256325393713251300528, 2.05198850934100499570405754287, 3.17088769461892826697290263688, 4.01776288179697908310705984438, 5.23273145975006303931368381943, 6.12038125562331313482140174900, 6.86431755830316686646475306592, 7.71968667740997823763187244701, 8.603033436243529957550609888822, 8.984813059232765598357530164127

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