↑

Properties

Label 2-1500-25.19-c1-0-6
Degree $2$
Conductor $1500$
Sign $0.737 - 0.675i$
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 − 1.57i·7-s + (−0.309 + 0.951i)9-s + (1.19 + 3.69i)11-s + (−0.326 − 0.106i)13-s + (3.56 − 4.91i)17-s + (2.98 + 2.17i)19-s + (1.27 − 0.928i)21-s + (−1.32 + 0.429i)23-s + (−0.951 + 0.309i)27-s + (−2.69 + 1.95i)29-s + (4.25 + 3.08i)31-s + (−2.28 + 3.13i)33-s + (8.14 + 2.64i)37-s + (−0.106 − 0.326i)39-s + ⋯
L(s)  = 1  + (0.339 + 0.467i)3-s − 0.596i·7-s + (−0.103 + 0.317i)9-s + (0.361 + 1.11i)11-s + (−0.0905 − 0.0294i)13-s + (0.865 − 1.19i)17-s + (0.685 + 0.497i)19-s + (0.278 − 0.202i)21-s + (−0.275 + 0.0895i)23-s + (−0.183 + 0.0594i)27-s + (−0.500 + 0.363i)29-s + (0.763 + 0.554i)31-s + (−0.397 + 0.546i)33-s + (1.33 + 0.435i)37-s + (−0.0169 − 0.0522i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.737 - 0.675i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.979124835\)
\(L(\frac12)\) \(\approx\) \(1.979124835\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.57iT - 7T^{2} \)
11 \( 1 + (-1.19 - 3.69i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.326 + 0.106i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.56 + 4.91i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.98 - 2.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.32 - 0.429i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.69 - 1.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.25 - 3.08i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.14 - 2.64i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.394 - 1.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.42iT - 43T^{2} \)
47 \( 1 + (0.220 + 0.303i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.64 - 9.14i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.57 - 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.38 + 10.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.14 + 8.46i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-8.19 + 5.95i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (12.5 - 4.07i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.1 + 8.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.71 - 3.74i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.24 - 6.91i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.55 + 4.89i)T + (-29.9 + 92.2i)T^{2} \)
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

−9.689883951291094787646192882291, −8.977252848808125001432487202016, −7.72286395337899813193821772407, −7.45586110433736994410406065815, −6.37906348375574291373471685063, −5.23778918022708727494161919394, −4.51655693119667132613999387668, −3.61097437173208671015353113340, −2.61638641835304504031945460484, −1.20437946797421111059769796066, 0.911804819337657184712872131131, 2.23894738466791678847471401317, 3.24173259899366291110297533605, 4.13032584284299172864183263127, 5.57114880839372819284923009931, 5.99447174067028510584402483435, 6.98615565603609295325750725758, 7.989715935718198039150912829752, 8.447305898126522358751015610556, 9.303795245405350232597534198241

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