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Properties

Label 2-1500-25.16-c1-0-3
Degree $2$
Conductor $1500$
Sign $-0.633 - 0.773i$
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  + (0.309 + 0.951i)3-s − 1.50·7-s + (−0.809 + 0.587i)9-s + (4.99 + 3.62i)11-s + (−2.87 + 2.09i)13-s + (−0.153 + 0.471i)17-s + (0.0963 − 0.296i)19-s + (−0.464 − 1.43i)21-s + (−2.47 − 1.79i)23-s + (−0.809 − 0.587i)27-s + (−0.0378 − 0.116i)29-s + (−0.909 + 2.79i)31-s + (−1.90 + 5.87i)33-s + (−3.53 + 2.56i)37-s + (−2.87 − 2.09i)39-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s − 0.568·7-s + (−0.269 + 0.195i)9-s + (1.50 + 1.09i)11-s + (−0.797 + 0.579i)13-s + (−0.0371 + 0.114i)17-s + (0.0220 − 0.0680i)19-s + (−0.101 − 0.312i)21-s + (−0.516 − 0.375i)23-s + (−0.155 − 0.113i)27-s + (−0.00701 − 0.0216i)29-s + (−0.163 + 0.502i)31-s + (−0.332 + 1.02i)33-s + (−0.580 + 0.421i)37-s + (−0.460 − 0.334i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254871693\)
\(L(\frac12)\) \(\approx\) \(1.254871693\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
good7 \( 1 + 1.50T + 7T^{2} \)
11 \( 1 + (-4.99 - 3.62i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.87 - 2.09i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.153 - 0.471i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.0963 + 0.296i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.47 + 1.79i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.0378 + 0.116i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.909 - 2.79i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.53 - 2.56i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.44 - 2.50i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.62T + 43T^{2} \)
47 \( 1 + (1.63 + 5.02i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.65 - 8.17i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.4 - 7.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.15 - 6.65i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.09 - 12.5i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.00 - 3.10i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-12.9 - 9.38i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.63 + 8.11i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.50 - 10.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.4 + 8.30i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.54 - 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.737496560903709140981914222271, −9.152111015476822854230914031076, −8.366482791254561757821464494180, −7.10016298326273363772575729362, −6.74418045904137732305156375478, −5.62922839200465139027380390750, −4.48983906615808062865259105136, −4.01273836005712034931849344248, −2.82046703832888159141445790091, −1.63521104290397592809130245616, 0.47818792459543622540384167445, 1.85915901309857914852578991369, 3.15644448278530738581727929901, 3.81007387355346628277811431999, 5.15574377459974113718244778177, 6.14319944847285388377297388140, 6.65088425094738304278998457938, 7.61683118005487009545606551124, 8.357776441666422116925124954571, 9.261886449043294447304113116149

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