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Properties

Label 2-1500-25.21-c1-0-11
Degree $2$
Conductor $1500$
Sign $-0.503 + 0.863i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.503 + 0.863i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9607512811\)
\(L(\frac12)\) \(\approx\) \(0.9607512811\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 \)
good7 \( 1 - 0.0883T + 7T^{2} \)
11 \( 1 + (-0.701 + 2.15i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.819 + 2.52i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.68 + 1.22i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.42 - 1.03i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.46 - 4.50i)T + (-18.6 - 13.5i)T^{2} \)
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 \( 1 + (2.19 + 6.77i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.03 + 6.26i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + (-8.17 - 5.94i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.777 + 0.565i)T + (16.3 + 50.4i)T^{2} \)
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 \( 1 + (-11.2 + 8.17i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.97 + 3.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.975 + 3.00i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 + 7.84i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.7 - 9.29i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.16 - 9.74i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.3 + 7.52i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.305719837960925380334396774294, −8.225615463604123519297668279862, −7.67147327129956328652515785861, −6.75701393227057606963416877095, −5.82320949978161794497219960174, −5.31624484601073243645254646492, −4.09091398548452227974048163031, −3.10497451460496251508334191358, −1.81504917980381258148855262306, −0.41496297961668520484592055850, 1.43206679605395727764877969940, 2.73017495324757775979535505987, 4.02668095098492491778772402277, 4.66258290251544325882647418628, 5.59473641793070380281138289031, 6.58603870149477170407239778926, 7.11912527772254318413925672264, 8.281922315215526058663543804825, 8.963572434846065553029767439938, 9.976262793090054151768124651511

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