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Properties

Label 2-1500-25.14-c1-0-5
Degree $2$
Conductor $1500$
Sign $0.922 - 0.385i$
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  + (−0.951 − 0.309i)3-s − 1.31i·7-s + (0.809 + 0.587i)9-s + (−1.25 + 0.913i)11-s + (−1.42 + 1.96i)13-s + (1.25 − 0.406i)17-s + (0.315 + 0.971i)19-s + (−0.407 + 1.25i)21-s + (2.94 + 4.05i)23-s + (−0.587 − 0.809i)27-s + (1.82 − 5.61i)29-s + (2.73 + 8.41i)31-s + (1.47 − 0.480i)33-s + (2.95 − 4.06i)37-s + (1.96 − 1.42i)39-s + ⋯
L(s)  = 1  + (−0.549 − 0.178i)3-s − 0.498i·7-s + (0.269 + 0.195i)9-s + (−0.379 + 0.275i)11-s + (−0.395 + 0.544i)13-s + (0.303 − 0.0985i)17-s + (0.0723 + 0.222i)19-s + (−0.0889 + 0.273i)21-s + (0.613 + 0.844i)23-s + (−0.113 − 0.155i)27-s + (0.338 − 1.04i)29-s + (0.491 + 1.51i)31-s + (0.257 − 0.0836i)33-s + (0.485 − 0.668i)37-s + (0.314 − 0.228i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252638877\)
\(L(\frac12)\) \(\approx\) \(1.252638877\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.31iT - 7T^{2} \)
11 \( 1 + (1.25 - 0.913i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.42 - 1.96i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.25 + 0.406i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.315 - 0.971i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.94 - 4.05i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.82 + 5.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.73 - 8.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.95 + 4.06i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.43 + 4.67i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + (-7.37 - 2.39i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.75 - 1.22i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.35 + 4.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.83 + 2.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.92 + 2.57i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.00 - 12.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.47 - 10.2i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.386 + 1.18i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.80 + 2.53i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-7.74 + 5.62i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-15.4 - 5.02i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.692937520623299455804709907557, −8.750464617523708680178249751141, −7.70818872028245504385230855374, −7.16344388956582123938030271692, −6.33000874395237071730661645871, −5.34958097766038047760475508672, −4.62022679105252838877689078305, −3.59652202701621689478848615212, −2.31436952383044084180290198612, −0.993895194906374773529545009853, 0.68693878214996598594666346524, 2.34660237524415193061145987723, 3.33021624888519966992128684433, 4.57789928055819979977960049302, 5.30277798487151269942284777069, 6.05727600560199735151791520956, 6.93546406027744848229591464121, 7.85777100136738171220252126666, 8.658908056377011479325999961476, 9.468110231512638874822137294864

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