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Properties

Label 2-1500-25.16-c1-0-9
Degree $2$
Conductor $1500$
Sign $0.994 - 0.101i$
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 − 3.54·7-s + (−0.809 + 0.587i)9-s + (1.78 + 1.29i)11-s + (5.80 − 4.21i)13-s + (1.96 − 6.05i)17-s + (−0.715 + 2.20i)19-s + (−1.09 − 3.37i)21-s + (−1.76 − 1.27i)23-s + (−0.809 − 0.587i)27-s + (0.262 + 0.806i)29-s + (−1.32 + 4.09i)31-s + (−0.681 + 2.09i)33-s + (5.84 − 4.24i)37-s + (5.80 + 4.21i)39-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s − 1.34·7-s + (−0.269 + 0.195i)9-s + (0.538 + 0.390i)11-s + (1.61 − 1.17i)13-s + (0.477 − 1.46i)17-s + (−0.164 + 0.504i)19-s + (−0.239 − 0.736i)21-s + (−0.367 − 0.266i)23-s + (−0.155 − 0.113i)27-s + (0.0486 + 0.149i)29-s + (−0.238 + 0.734i)31-s + (−0.118 + 0.365i)33-s + (0.961 − 0.698i)37-s + (0.929 + 0.675i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.994 - 0.101i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.994 - 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642446094\)
\(L(\frac12)\) \(\approx\) \(1.642446094\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 + (-1.78 - 1.29i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-5.80 + 4.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.96 + 6.05i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.715 - 2.20i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.76 + 1.27i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.262 - 0.806i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.32 - 4.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.84 + 4.24i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.08 + 0.790i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.18T + 43T^{2} \)
47 \( 1 + (-1.87 - 5.75i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.69 - 11.3i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.0 + 7.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.59 - 4.06i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.46 + 4.50i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-4.25 - 13.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.881 + 0.640i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.80 + 5.55i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.87 + 11.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.68 - 4.12i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.60 + 17.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.454826453093856382044160340407, −8.931243150633465335645396889543, −7.955208001763802058159752489246, −7.05069849654000954344596552348, −6.09581789735545171749144177663, −5.53627804929435420003905715634, −4.21233141882572948828779380466, −3.46292856312444965303922414498, −2.69954690118838042985104430695, −0.846201857012956014838616478782, 1.01920757532477663892016985960, 2.26687706943563066325076175860, 3.65100471083357298207255957176, 3.93651345260901501029124275227, 5.72955824131271885867939469627, 6.36656509578269869138427672515, 6.74838781353583552215827227198, 7.973335964588941945383527510780, 8.713089443892948464228212493928, 9.329815918501584439339396873488

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